Inmathematics, thespin representations are particularprojective representations of theorthogonal orspecial orthogonal groups in arbitrarydimension andsignature (i.e., includingindefinite orthogonal groups). More precisely, they are two equivalentrepresentations of thespin groups, which aredouble covers of the special orthogonal groups. They are usually studied over thereal orcomplex numbers, but they can be defined over otherfields.

Elements of a spin representation are calledspinors. They play an important role in thephysical description offermions such as theelectron.

The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximalisotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derivereal representations by introducingreal structures.

The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admitinvariantbilinear forms, which can be used toembed the spin groups intoclassical Lie groups. In low dimensions, these embeddings aresurjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.

LetV be afinite-dimensional real or complexvector space with anondegeneratequadratic formQ. The (real or complex)linear maps preservingQ form theorthogonal groupO(V,Q). Theidentity component of the group is called the special orthogonal groupSO(V,Q). (ForV real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up togroup isomorphism,SO(V,Q) has a uniqueconnecteddouble cover, the spin groupSpin(V,Q). There is thus agroup homomorphismh: Spin(V,Q) → SO(V,Q) whosekernel has two elements denoted{1, −1}, where1 is theidentity element. Thus, the group elementsg and−g ofSpin(V,Q) are equivalent after the homomorphism toSO(V,Q); that is,h(g) =h(−g) for anyg inSpin(V,Q).

The groupsO(V,Q), SO(V,Q) andSpin(V,Q) are allLie groups, and for fixed(V,Q) they have the sameLie algebra,so(V,Q). IfV is real, thenV is a real vector subspace of itscomplexificationV_C =V ⊗_RC, and the quadratic formQ extends naturally to a quadratic formQ_C onV_C. This embedsSO(V,Q) as asubgroup ofSO(V_C,Q_C), and hence we may realiseSpin(V,Q) as a subgroup ofSpin(V_C,Q_C). Furthermore,so(V_C,Q_C) is the complexification ofso(V,Q).

In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimensionn ofV. Concretely, we may assumeV =Cⁿ and

${\displaystyle Q(z_1,\dots ,z_n)=z_1^2+z_2^2+\cdots +z_n^2.}$

The corresponding Lie groups are denotedO(n,C), SO(n,C), Spin(n,C) and their Lie algebra asso(n,C).

In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers(p,q) wheren =p +q is the dimension ofV, andp −q is thesignature. Concretely, we may assumeV =Rⁿ and

${\displaystyle Q(x_1,\dots ,x_n)=x_1^2+x_2^2+\cdots +x_p^2-(x_{p+1}^2+\cdots +x_{p+q}^2).}$

The corresponding Lie groups and Lie algebra are denotedO(p,q), SO(p,q), Spin(p,q) andso(p,q). We writeR^p,q in place ofRⁿ to make the signature explicit.

The spin representations are, in a sense, the simplestrepresentations ofSpin(n,C) andSpin(p,q) that do not come from representations ofSO(n,C) andSO(p,q). A spin representation is, therefore, a real or complex vector spaceS together with a group homomorphismρ fromSpin(n,C) orSpin(p,q) to thegeneral linear groupGL(S) such that the element−1 isnot in the kernel ofρ.

IfS is such a representation, then according to the relation between Lie groups and Lie algebras, it induces aLie algebra representation, i.e., aLie algebra homomorphism fromso(n,C) orso(p,q) to the Lie algebragl(S) ofendomorphisms ofS with thecommutator bracket.

Spin representations can be analysed according to the following strategy: ifS is a real spin representation ofSpin(p,q), then its complexification is a complex spin representation ofSpin(p,q); as a representation ofso(p,q), it therefore extends to a complex representation ofso(n,C). Proceeding in reverse, we thereforefirst construct complex spin representations ofSpin(n,C) andso(n,C), then restrict them to complex spin representations ofso(p,q) andSpin(p,q), then finally analyse possible reductions to real spin representations.

LetV =Cⁿ with the standard quadratic formQ so that

${\displaystyle \mathfrak{s}\mathfrak{o}(V,Q)=\mathfrak{s}\mathfrak{o}(n,\mathbb{C}).}$

Thesymmetric bilinear form onV associated toQ bypolarization is denoted⟨.,.⟩.

Suppose that for${\displaystyle x,y\in V}$. Then${\displaystyle Q(x+iy)=Q(x)-Q(y)=0}$. We say that${\displaystyle x+iy}$ is isotropic and that it generates a one-dimensional totally isotropic subspace${\displaystyle \{\lambda (x+iy)|\lambda \in C\}}$. Similary,${\displaystyle x-iy}$ is isotropic and generates a one-dimensional totally isotropic subspace.

A standard construction of the spin representations ofso(n,C) begins with a choice of a pair(W,W^∗)of maximaltotally isotropic subspaces (with respect toQ) ofV withW ∩W^∗ = 0. Let us make such a choice. Ifn = 2m orn = 2m + 1, thenW andW^∗ both have dimensionm. Ifn = 2m, thenV =W ⊕W^∗, whereas ifn = 2m + 1, thenV =W ⊕U ⊕W^∗, whereU is the 1-dimensional orthogonal complement toW ⊕W^∗. The bilinear form⟨.,.⟩ associated toQ induces apairing betweenW andW^∗, which must be nondegenerate, becauseW andW^∗ are totally isotropic subspaces andQ is nondegenerate. HenceW andW^∗ aredual vector spaces.

More concretely, leta₁, ...a_m be a basis forW. Then there is a unique basisα₁, ...α_m ofW^∗ such that

${\displaystyle ⟨{\alpha }_i,a_j⟩={\delta }_{ij}.}$

IfA is anm ×m matrix, thenA induces an endomorphism ofW with respect to this basis and the transposeA^T induces a transformation ofW^∗ with

${\displaystyle ⟨Aw,w^{\ast }⟩=⟨w,A^{\mathrm{T}}{w}^{\ast }⟩}$

for allw inW andw^∗ inW^∗. It follows that the endomorphismρ_A ofV, equal toA onW,−A^T onW^∗ and zero onU (ifn is odd), is skew,

${\displaystyle ⟨{\rho }_{A}u,v⟩=-⟨u,{\rho }_{A}v⟩}$

for allu,v inV, and hence (seeclassical group) an element ofso(n,C) ⊂ End(V).

Using the diagonal matrices in this construction defines aCartan subalgebrah ofso(n,C): therank ofso(n,C) ism, and the diagonaln ×n matrices determine anm-dimensional abelian subalgebra.

Letε₁, ...ε_m be the basis ofh^∗ such that, for a diagonal matrixA,ε_k(ρ_A) is thekth diagonal entry ofA. Clearly this is a basis forh^∗. Since the bilinear form identifiesso(n,C) with${\displaystyle {\wedge }^{2}V}$, explicitly,

${\displaystyle x\wedge y\mapsto {\phi }_{x\wedge y},{\phi }_{x\wedge y}(v)=⟨y,v⟩x-⟨x,v⟩y,x\wedge y\in {\wedge }^{2}V,x,y,v\in V,{\phi }_{x\wedge y}\in \mathfrak{s}\mathfrak{o}(n,\mathbb{C}),}$^[1]

it is now easy to construct theroot system associated toh. Theroot spaces (simultaneous eigenspaces for the action ofh) are spanned by the following elements:

${\displaystyle a_i\wedge a_j,i\ne j,}$ withroot (simultaneous eigenvalue)${\displaystyle {\epsilon }_i+{\epsilon }_j}$

${\displaystyle a_i\wedge {\alpha }_j}$ (which is inh ifi =j) with root${\displaystyle {\epsilon }_i-{\epsilon }_j}$

${\displaystyle {\alpha }_i\wedge {\alpha }_j,i\ne j,}$ with root${\displaystyle -{\epsilon }_i-{\epsilon }_j,}$

and, ifn is odd, andu is a nonzero element ofU,

${\displaystyle a_i\wedge u,}$ with root${\displaystyle {\epsilon }_i}$

${\displaystyle {\alpha }_i\wedge u,}$ with root${\displaystyle -{\epsilon }_i.}$

Thus, with respect to the basisε₁, ...ε_m, the roots are the vectors inh^∗ that are permutations of

${\displaystyle (\pm 1,\pm 1,0,0,\dots ,0)}$

together with the permutations of

${\displaystyle (\pm 1,0,0,\dots ,0)}$

ifn = 2m + 1 is odd.

A system ofpositive roots is given byε_i +ε_j (i ≠j),ε_i −ε_j (i <j) and (forn odd)ε_i. The correspondingsimple roots are

The positive roots are nonnegative integer linear combinations of the simple roots.

One construction of the spin representations ofso(n,C) uses theexterior algebra(s)

${\displaystyle S={\wedge }^{\bullet }W}$ and/or${\displaystyle S^{\prime }={\wedge }^{\bullet }{W}^{\ast }.}$

There is an action ofV onS such that for any elementv =w +w^∗ inW ⊕W^∗ and anyψ inS the action is given by:

${\displaystyle v\cdot \psi =2^{\frac{1}{2}}(w\wedge \psi +\iota (w^{\ast })\psi ),}$

where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairsW andW^∗. This action respects theClifford relationsv² =Q(v)1, and so induces a homomorphism from theClifford algebraCl_nC ofV toEnd(S). A similar action can be defined onS′, so that bothS andS′ areClifford modules.

The Lie algebraso(n,C) is isomorphic to the complexified Lie algebraspin_n^C inCl_nC via the mapping induced by the coveringSpin(n) → SO(n)^[2]

${\displaystyle v\wedge w\mapsto \frac{1}{4}[v,w].}$

It follows that bothS andS′ are representations ofso(n,C). They are actuallyequivalent representations, so we focus onS.

The explicit description shows that the elementsα_i ∧a_i of the Cartan subalgebrah act onS by

${\displaystyle ({\alpha }_i\wedge a_i)\cdot \psi =\frac{1}{4}(2^{\frac{1}{2}}{)}^2(\iota ({\alpha }_i)(a_i\wedge \psi )-a_i\wedge (\iota ({\alpha }_i)\psi ))=\frac{1}{2}\psi -a_i\wedge (\iota ({\alpha }_i)\psi ).}$

A basis forS is given by elements of the form

${\displaystyle a_{i_1}\wedge a_{i_2}\wedge \cdots \wedge a_{i_k}}$

for0 ≤k ≤m andi₁ < ... <i_k. These clearly spanweight spaces for the action ofh:α_i ∧a_i has eigenvalue −1/2 on the given basis vector ifi =i_j for somej, and has eigenvalue1/2 otherwise.

It follows that theweights ofS are all possible combinations of

${\displaystyle (\pm \frac{1}{2},\pm \frac{1}{2},\dots \pm \frac{1}{2})}$

and eachweight space is one-dimensional. Elements ofS are calledDirac spinors.

Whenn is even,S is not anirreducible representation:${\displaystyle S_{+}={\wedge }^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}W}$ and${\displaystyle S_{-}={\wedge }^{\mathrm{o}\mathrm{d}\mathrm{d}}W}$ are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. BothS₊ andS₋ are irreducible representations of dimension 2^m−1 whose elements are calledWeyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, thehighest weights ofS₊ andS₋ are

${\displaystyle (\frac{1}{2},\frac{1}{2},\dots \frac{1}{2},\frac{1}{2})}$ and${\displaystyle (\frac{1}{2},\frac{1}{2},\dots \frac{1}{2},-\frac{1}{2})}$

respectively. The Clifford action identifies Cl_nC with End(S) and theeven subalgebra is identified with the endomorphisms preservingS₊ andS₋. The otherClifford moduleS′ isisomorphic toS in this case.

Whenn is odd,S is an irreducible representation ofso(n,C) of dimension 2^m: the Clifford action of a unit vectoru ∈U is given by

and so elements ofso(n,C) of the formu∧w oru∧w^∗ do not preserve the even and odd parts of the exterior algebra ofW. The highest weight ofS is

${\displaystyle (\frac{1}{2},\frac{1}{2},\dots \frac{1}{2}).}$

The Clifford action is not faithful onS: Cl_nC can be identified with End(S) ⊕ End(S′), whereu acts with the opposite sign onS′. More precisely, the two representations are related by theparityinvolutionα of Cl_nC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl_nC. In other words, there is alinear isomorphism fromS toS′, which identifies the action ofA in Cl_nC onS with the action ofα(A) onS′.

ifλ is a weight ofS, so is −λ. It follows thatS is isomorphic to thedual representationS^∗.

Whenn = 2m + 1 is odd, the isomorphismB:S →S^∗ is unique up to scale bySchur's lemma, sinceS is irreducible, and it defines a nondegenerate invariant bilinear formβ onS via

${\displaystyle \beta (\phi ,\psi )=B(\phi )(\psi ).}$

Here invariance means that

${\displaystyle \beta (\xi \cdot \phi ,\psi )+\beta (\phi ,\xi \cdot \psi )=0}$

for allξ inso(n,C) andφ,ψ inS — in other words the action ofξ is skew with respect toβ. In fact, more is true:S^∗ is a representation of theopposite Clifford algebra, and therefore, since Cl_nC only has two nontrivialsimple modulesS andS′, related by the parity involutionα, there is anantiautomorphismτ of Cl_nC such that

${\displaystyle \beta (A\cdot \phi ,\psi )=\beta (\phi ,\tau (A)\cdot \psi )(1)}$

for anyA in Cl_nC. In factτ is reversion (the antiautomorphism induced by the identity onV) form even, and conjugation (the antiautomorphism induced by minus the identity onV) form odd. These two antiautomorphisms are related by parity involutionα, which is the automorphism induced by minus the identity onV. Both satisfyτ(ξ) = −ξ forξ inso(n,C).

Whenn = 2m, the situation depends more sensitively upon the parity ofm. Form even, a weightλ has an even number of minus signs if and only if −λ does; it follows that there are separate isomorphismsB_±:S_± →S_±^∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphismB:S →S^∗. Form odd,λ is a weight ofS₊ if and only if −λ is a weight ofS₋; thus there is an isomorphism fromS₊ toS₋^∗, again unique up to scale, and itstranspose provides an isomorphism fromS₋ toS₊^∗. These may again be combined into an isomorphismB:S →S^∗.

For bothm even andm odd, the freedom in the choice ofB may be restricted to an overall scale by insisting that the bilinear formβ corresponding toB satisfies (1), whereτ is a fixed antiautomorphism (either reversion or conjugation).

The symmetry properties ofβ:S ⊗S →C can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor squareS ⊗S must decompose into a direct sum ofk-forms onV for variousk, because its weights are all elements inh^∗ whose components belong to {−1,0,1}. Nowequivariant linear mapsS ⊗S → ∧^kV^∗ correspond bijectively to invariant maps ∧^kV ⊗S ⊗S →C and nonzero such maps can be constructed via the inclusion of ∧^kV into the Clifford algebra. Furthermore, ifβ(φ,ψ) =εβ(ψ,φ) andτ has signε_k on ∧^kV then

${\displaystyle \beta (A\cdot \phi ,\psi )=\epsilon {\epsilon }_k\beta (A\cdot \psi ,\phi )}$

forA in ∧^kV.

Ifn = 2m+1 is odd then it follows from Schur's Lemma that

${\displaystyle S\otimes S\cong \underset{j=0}{\overset{m}{⨁}}{\wedge }^{2j}{V}^{\ast }}$

(both sides have dimension 2^2m and the representations on the right are inequivalent). Because the symmetries are governed by an involutionτ that is either conjugation or reversion, the symmetry of the ∧^2jV^∗ component alternates withj. Elementary combinatorics gives

${\displaystyle \sum _{j=0}^m(-1{)}^j\dim {\wedge }^{2j}{\mathbb{C}}^{2m+1}=(-1{)}^{\frac{1}{2}m(m+1)}{2}^m=(-1{)}^{\frac{1}{2}m(m+1)}(\dim {\mathrm{S}}^{2}S-\dim {\wedge }^{2}S)}$

and the sign determines which representations occur in S²S and which occur in ∧²S.^[3] In particular

${\displaystyle \beta (\varphi ,\psi )=(-1{)}^{\frac{1}{2}m(m+1)}\beta (\psi ,\varphi ),}$ and

${\displaystyle \beta (v\cdot \varphi ,\psi )=(-1{)}^m(-1{)}^{\frac{1}{2}m(m+1)}\beta (v\cdot \psi ,\varphi )=(-1{)}^m\beta (\varphi ,v\cdot \psi )}$

forv ∈V (which is isomorphic to ∧^2mV), confirming thatτ is reversion form even, and conjugation form odd.

Ifn = 2m is even, then the analysis is more involved, but the result is a more refined decomposition: S²S_±, ∧²S_± andS₊ ⊗S₋ can each be decomposed as a direct sum ofk-forms (where fork =m there is a further decomposition into selfdual and antiselfdualm-forms).

The main outcome is a realisation ofso(n,C) as a subalgebra of a classical Lie algebra onS, depending uponn modulo 8, according to the following table:

n mod 8	0	1	2	3	4	5	6	7
Spinor algebra	${\displaystyle \mathfrak{s}\mathfrak{o}(S_{+})\oplus \mathfrak{s}\mathfrak{o}(S_{-})}$	${\displaystyle \mathfrak{s}\mathfrak{o}(S)}$	${\displaystyle \mathfrak{g}\mathfrak{l}(S_{\pm })}$	${\displaystyle \mathfrak{s}\mathfrak{p}(S)}$	${\displaystyle \mathfrak{s}\mathfrak{p}(S_{+})\oplus \mathfrak{s}\mathfrak{p}(S_{-})}$	${\displaystyle \mathfrak{s}\mathfrak{p}(S)}$	${\displaystyle \mathfrak{g}\mathfrak{l}(S_{\pm })}$	${\displaystyle \mathfrak{s}\mathfrak{o}(S)}$

Forn ≤ 6, these embeddings are isomorphisms (ontosl rather thangl forn = 6):

${\displaystyle \mathfrak{s}\mathfrak{o}(2,\mathbb{C})\cong \mathfrak{g}\mathfrak{l}(1,\mathbb{C})(=\mathbb{C})}$

${\displaystyle \mathfrak{s}\mathfrak{o}(3,\mathbb{C})\cong \mathfrak{s}\mathfrak{p}(2,\mathbb{C})(=\mathfrak{s}\mathfrak{l}(2,\mathbb{C}))}$

${\displaystyle \mathfrak{s}\mathfrak{o}(4,\mathbb{C})\cong \mathfrak{s}\mathfrak{p}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{p}(2,\mathbb{C})}$

${\displaystyle \mathfrak{s}\mathfrak{o}(5,\mathbb{C})\cong \mathfrak{s}\mathfrak{p}(4,\mathbb{C})}$

${\displaystyle \mathfrak{s}\mathfrak{o}(6,\mathbb{C})\cong \mathfrak{s}\mathfrak{l}(4,\mathbb{C}).}$

The complex spin representations ofso(n,C) yield real representationsS ofso(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.

1. There is an invariant complex antilinear mapr:S →S withr² = id_S. The fixed point set ofr is then a real vector subspaceS_R ofS withS_R ⊗C =S. This is called areal structure.
2. There is an invariant complex antilinear mapj:S →S withj² = −id_S. It follows that the triplei,j andk:=ij makeS into a quaternionic vector spaceS_H. This is called aquaternionic structure.
3. There is an invariant complex antilinear mapb:S →S^∗ that is invertible. This defines a pseudohermitian bilinear form onS and is called ahermitian structure.

The type of structure invariant underso(p,q) depends only on the signaturep −q modulo 8, and is given by the following table.

HereR,C andH denote real, hermitian and quaternionic structures respectively, andR +R andH +H indicate that the half-spin representations both admit real or quaternionic structures respectively.

To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Sincen =p +q ≅p −q mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.

The odd case is simpler, there is only one complex spin representationS, and hermitian structures do not occur. Apart from the trivial casen = 1,S is always even-dimensional, say dimS = 2N. The real forms ofso(2N,C) areso(K,L) withK +L = 2N andso^∗(N,H), while the real forms ofsp(2N,C) aresp(2N,R) andsp(K,L) withK +L =N. The presence of a Clifford action ofV onS forcesK =L in both cases unlesspq = 0, in which caseKL=0, which is denoted simplyso(2N) orsp(N). Hence the odd spin representations may be summarized in the following table.

(†)N is even forn > 3 and forn = 3, this issp(1).

The even-dimensional case is similar. Forn > 2, the complex half-spin representations are even-dimensional. We have additionally to deal with hermitian structures and the real forms ofsl(2N,C), which aresl(2N,R),su(K,L) withK +L = 2N, andsl(N,H). The resulting even spin representations are summarized as follows.

(*) Forpq = 0, we have insteadso(2N) +so(2N)

(†)N is even forn > 4 and forpq = 0 (which includesn = 4 withN = 1), we have insteadsp(N) +sp(N)

The low-dimensional isomorphisms in the complex case have the following real forms.

Euclidean signature	Minkowskian signature	Other signatures
${\displaystyle \mathfrak{s}\mathfrak{o}(2)\cong \mathfrak{u}(1)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(1,1)\cong \mathbb{R}}$
${\displaystyle \mathfrak{s}\mathfrak{o}(3)\cong \mathfrak{s}\mathfrak{p}(1)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(2,1)\cong \mathfrak{s}\mathfrak{l}(2,\mathbb{R})}$
${\displaystyle \mathfrak{s}\mathfrak{o}(4)\cong \mathfrak{s}\mathfrak{p}(1)\oplus \mathfrak{s}\mathfrak{p}(1)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(3,1)\cong \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$	${\displaystyle \mathfrak{s}\mathfrak{o}(2,2)\cong \mathfrak{s}\mathfrak{l}(2,\mathbb{R})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{R})}$
${\displaystyle \mathfrak{s}\mathfrak{o}(5)\cong \mathfrak{s}\mathfrak{p}(2)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(4,1)\cong \mathfrak{s}\mathfrak{p}(1,1)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(3,2)\cong \mathfrak{s}\mathfrak{p}(4,\mathbb{R})}$
${\displaystyle \mathfrak{s}\mathfrak{o}(6)\cong \mathfrak{s}\mathfrak{u}(4)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(5,1)\cong \mathfrak{s}\mathfrak{l}(2,\mathbb{H})}$	${\displaystyle \mathfrak{s}\mathfrak{o}(4,2)\cong \mathfrak{s}\mathfrak{u}(2,2)}$	${\displaystyle \mathfrak{s}\mathfrak{o}(3,3)\cong \mathfrak{s}\mathfrak{l}(4,\mathbb{R})}$

The only special isomorphisms of real Lie algebras missing from this table are${\displaystyle {\mathfrak{s}\mathfrak{o}}^{\ast }(3,\mathbb{H})\cong \mathfrak{s}\mathfrak{u}(3,1)}$ and${\displaystyle {\mathfrak{s}\mathfrak{o}}^{\ast }(4,\mathbb{H})\cong \mathfrak{s}\mathfrak{o}(6,2).}$

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