First of all, I'm from a physics background, so pardon my probable lack of mathematical rigor in my question.
Let's say I have an 8-dimensional Clifford algebra $C\ell_8$ with generators $e_i$, for which I choose some 16-dimensional matrix representation. I can write a vector in this algebra as a $16 \times 16$ matrix $X=x_i e_i$.
Now, I'm aware that spinors in $C\ell_8$ are 16-dimensional vectors with 8 non-zero components and that there are 2 types of spinors. Triality, as it appears to me, relates these 3 vector spaces.
The point where I'm struggling with is to understand explicitly what triality really implies in terms of matrices in $C\ell_8$. I've read that it implies all 3 representations are equivalent so in some sense $X = S Z S^{-1}$, where $Z$ is some matrix representation of a spinor and $S$ some other matrix?
Thanks for any help!