Triality and central quotients of Spin(8)

Statement

For the compact real groups, triality gives

\operatorname{Out}(\operatorname{Spin}(8))\cong S_3, \qquad \operatorname{Out}(SO(8))\cong C_2.

The center of $\operatorname{Spin}(8)$ is $C_2\times C_2$ . Its three order-two subgroups are the distinct kernels of $8_v,8_+,8_-$ ; the triality $S_3$ permutes them. Only the subgroup fixing the vector kernel descends to automorphisms of the fixed quotient $SO(8)$ .

Descent criterion

An automorphism $\phi:G\to G$ induces an automorphism of $G/K$ precisely when $\phi(K)=K$ . The stabilizer of one item in a three-element permutation action is $C_2$ . Thus an order-three triality automorphism does not descend to this quotient.

Lie algebra level

The compact real Lie algebra $\mathfrak{so}(8)$ retains outer automorphism group $S_3$ . Differentiation forgets the central kernel of a covering, so a Lie-algebra symmetry need not integrate to an automorphism of every global group with that Lie algebra.

References

Laura P. Schaposnik and Sebastian Schulz, “Triality for Homogeneous Polynomials,” §2.2, Proposition 2.1 and Remark 2.2, p. 5. Paper.