Let $G$ be a Lie group. The set of all $N$ such that $G$ is a subgroup of $SO(N)$ has a minimum $N_{\min}(G)$. (If I am not wrong, $N_{\min}(G)$ is supposed to be less or equal to $\dim(G)$)
What is the value of $N_{\min}(G)$, with $G$ = (compact real forms) $E_6, E_7, E_8, E_6\times E_6, E_7\times E_7, E_8\times E_8$ ?