The vector representation of Spin(8)

The vector representation $8_v$ of compact $\operatorname{Spin}(8)$ is the real space $\mathbb R^8$ with action obtained from the canonical double cover

\rho_v:\operatorname{Spin}(8)\longrightarrow SO(8)

and the usual matrix action. Its dimension is $8$ , it is irreducible over $\mathbb R$ , and $\ker\rho_v=\{1,-1\}$ .

The action is part of the object

The positive and negative half-spin representations also have real dimension eight, but have different actions and central kernels. They are distinct objects in the category of representations of this fixed group. Triality permutes their isomorphism classes by twisting the group action.

References

John C. Baez, “The Octonions,” §2.4, vector representations and Table 4. Checked section.