Among all special orthogonal groups SO(n)SO(n), the case of SO(8)SO(8) is special, since in the ADE classification of simple Lie groups it corresponds to D4, which makes its representation theory enjoy triality.


Subgroup lattice


(Spin(7)-subgroups in Spin(8))

There are precisely 3 conjugacy classes of Spin(7)-subgroups inside Spin(8), and the triality group Out(Spin(8))Out(Spin(8)) acts transitively on these three classes.

(Varadarajan 01, Theorem 5 on p. 6, see also Kollross 02, Prop. 3.3 (1))


(G2 is intersection of Spin(7)-subgroups of Spin(8))

The intersection inside Spin(8) of any two Spin(7)-subgroups from distinct conjugacy classes of subgroups (according to Prop. ) is the exceptional Lie group G2, hence we have pullback squares of the form

(Varadarajan 01, Theorem 5 on p. 13)


We have the following commuting diagram of subgroup inclusions, where each square exhibits a pullback/fiber product, hence an intersection of subgroups:

Here in the bottom row we have the Lie groups

Spin(5)\hookrightarrow Spin(6) \hookrightarrow Spin(7) \hookrightarrow Spin(8)

with their canonical subgroup-inclusions, while in the top row we have

SU(2)\hookrightarrow SU(3) \hookrightarrow G2 \hookrightarrow Spin(7)

and the right vertical inclusion BιB \iota' is the delooping of one of the two non-standard inclusions, according to Prop. .


The square on the right is that from Prop. .

The square in the middle is Varadarajan 01, Lemma 9 on p. 10.

The statement also follows with Onishchik 93, Table 2, p. 144:


(Spin(5).Spin(3)-subgroups in SO(8))

The direct product group SO(3) ×\times SO(5) together with the groups Sp(2).Sp(1) and Sp(1)Sp(2)Sp(1) \cdot Sp(2), with their canonical inclusions into SO(8), form 3 conjugacy classes of subgroups inside SO(8), and the triality group Out(Spin(8))Out(Spin(8)) acts transitively on these three classes.

(Kollross 02, Prop. 3.3 (3))



(Spin(5).Spin(3)-subgroups in Spin(8))

The groups Spin(5).Spin(3), Sp(2).Sp(1) and Sp(1)Sp(2)Sp(1) \cdot Sp(2), with their canonical inclusions into Spin(8), form 3 conjugacy classes of subgroups inside Spin(8), and the triality group Out(Spin(8))Out(Spin(8)) acts transitively on these three classes.

(Čadek-Vanžura 97, Sec. 2)

In summary we have these subgroup inclusions

permuted by triality:

graphics grabbed from FSS 19, Sec. 3.3

Homotopy groups

The homotopy groups of SO(8)SO(8) in low degrees:

GGπ 1\pi_1π 2\pi_2π 3\pi_3π 4\pi_4π 5\pi_5π 6\pi_6π 7\pi_7π 8\pi_8π 9\pi_9π 10\pi_10π 11\pi_11π 12\pi_12π 13\pi_13π 14\pi_14π 15\pi_15
SO(8)SO(8) 2\mathbb{Z}_200\mathbb{Z}000000 2\mathbb{Z}^{\oplus 2} 2 3\mathbb{Z}_{2}^{\oplus 3} 2 3\mathbb{Z}_{2}^{\oplus 3} 8 24\mathbb{Z}_{8} \oplus \mathbb{Z}_{24} 2\mathbb{Z}_2 \oplus \mathbb{Z}0 2\mathbb{Z}^{\oplus 2} 2 8 120 2520\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520} 2 7\mathbb{Z}_2^{\oplus 7}

Cohomology of classifying spaces


The ordinary cohomology ring of the classifying space BSpin(8)B Spin(8) is:

1) with coefficients in the cyclic group of order 2:

H (BSpin(8), 2)[w 4,w 6,w 7,w 8,ρ 2(14(p 2(12p 1) 22χ))] H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \;\simeq\; \mathbb{Z} \big[ w_4, w_6, w_7, w_8, \; \rho_2 \left( \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) \right) \big]

where w iw_i are the universal Stiefel-Whitney classes,

and where

ρ 2:H (BSpin(8),)H (BSpin(8), 2) \rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)

is mod 2 reduction

2) with coefficients in the integers:

H (BSpin(8),)[12p 1,14(p 2(12p 1) 2)12χ,χ,β(w 6)]/2β(w 6), H^\bullet \big( B Spin(8), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \Big[ \tfrac{1}{2}p_1, \; \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \right) - \tfrac{1}{2}\chi , \; \chi, \; \beta(w_6) \Big] / \big\langle 2 \beta(w_6)\big\rangle \,,

where p 1p_1 is the first fractional Pontryagin class, p 2p_2 is the second Pontryagin class, χ\chi is the Euler class, and

β:H (BSpin(8), 2)H +1(BSpin(8),) \beta \;\colon\; H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \longrightarrow H^{\bullet + 1} \big( B Spin(8), \mathbb{Z} \big)

is the Bockstein homomorphism.

Moreover, we have the following relations:

ρ 2(12p 1) =w 4 ρ 2(χ) =w 8 \begin{aligned} \rho_2\left( \tfrac{1}{2}p_1 \right) & = w_4 \\ \rho_2\big( \chi\big) & = w_8 \end{aligned}

This is due to Quillen 71, Čadek-Vanžura 95, see Čadek-Vanžura 97, Lemma 4.1.


Consider the delooping of the triality automorphism relating Sp(2).Sp(1) with Spin(5).Spin(3) (Prop. ) on classifying spaces

B(Spin(5)Spin(3)) BSpin(8) Btri B(Sp(2)Sp(1)) BSpin(8) \array{ B \big( Spin(5) \cdot Spin(3) \big) &\hookrightarrow& B Spin(8) \\ \big\downarrow && \big\downarrow^{ B \mathrlap{tri} } \\ B \big( Sp(2) \cdot Sp(1) \big) &\hookrightarrow& B Spin(8) }

Then the pullback of the universal characteristic classes of BSpin(8)B Spin(8) (from Prop. ) along BtriB tri is as follows:

(Btri) *:12p 1 12p 1 χ 14(p 2(12p 1) 2)+12χ 14(p 2(12p 1) 2)12χ χ \big( B tri \big)^\ast \;\colon\; \begin{aligned} \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1 \\ \chi & \mapsto - \tfrac{1}{4} \big( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \big) + \tfrac{1}{2}\chi \\ \tfrac{1}{4} \big( p_2 - \big(\tfrac{1}{2}p_1\big)^2 \big) - \tfrac{1}{2}\chi & \mapsto - \chi \end{aligned}

(Čadek-Vanžura 97, Lemma 4.2)

In fact tri 1=tritri^{-1} = tri.

Hence, in rational cohomology:

(Btri) *(14p 2) =(Btri) *((14p 2(14p 1) 212χ)+(14p 1) 2+12χ) =χ+(14p 1) 212(14p 2(14p 1) 212χ) \begin{aligned} \big( B tri \big)^\ast \big( \tfrac{1}{4}p_2 \big) & = \big( B tri \big)^\ast \Big( \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) + \big(\tfrac{1}{4}p_1\big)^2 + \tfrac{1}{2}\chi \Big) \\ & = -\chi + \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2} \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) \end{aligned}

GG-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group

see also



See also

Subgroup lattice

On the subgroup lattice of Spin(8)

  • A. L. Onishchik (ed.) Lie Groups and Lie Algebras

    • I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

    • II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

    Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

  • Veeravalli Varadarajan, Spin(7)-subgroups of SO(8) and Spin(8), Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, pdf)

  • Martin Čadek, Jiří Vanžura, On Sp(2)Sp(2) and Sp(2)Sp(1)Sp(2) \cdot Sp(1)-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

  • Megan M. Kerr, New examples of homogeneous Einstein metrics, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (euclid:1030132086)

  • Andreas Kollross, Prop. 3.3 of A Classification of Hyperpolar and Cohomogeneity One Actions, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)

Discussion with an eye towards foundations of M-theory:


The integral cohomology of the classifying spaces BSO(8)B SO(8) and BSpin(8)B Spin(8) and the action of triality on these is discussed in

  • Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)

  • Daniel Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann . 194 (1971), 19

  • Martin Čadek, Jiří Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 (dml-cz:118764)

  • Martin Čadek, Jiří Vanžura, Section 2 of On Sp(2)Sp(2) and Sp(2)Sp(1)Sp(2) \cdot Sp(1)-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

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