This post is a follow-up to my earlier post on matrices for classical groups.

I wanted to do an investigation of the triality automorphism of the orthogonal group $O_8^+(q)$. I finally got around to this today, and have written up some rough notes which are here. One unexpected bonus was that I was able to write down matrices for the natural 8-dimensional representation of $G_2(q)$ over $\mathbb{F}{q}$ and ${^3D_4}(q)$ over $\mathbb{F}_{q^3}$.

Both of these families of exceptional groups lie inside $O_8^+$ groups. Their structure, when written as matrices, is surprisingly similar (surprising to me!). Although all of this is well-known to experts, I’ve not seen the matrices written down before so I was pleased that I could nut it out…