Symmetries on the real form $\mathfrak e_{6,-26}$

Abstract

We describe four fine gradings on the real form $\mathfrak e_{6,-26}$. They are precisely the gradings whose complexifications are fine gradings on the complexified algebra $\mathfrak{e}_6$. The universal grading groups are $\mathbb Z_2^6$, $\mathbb Z\times\mathbb Z_2^4$, $\mathbb Z^2\times\mathbb Z_2^3$, and $\mathbb Z_4\times\mathbb Z_2^4$.