  Let $G$ be a connected closed subgroup of $\mathrm{GL}_n(\mathbb{C})$ which
is simple as a Lie group and which acts irreducibly on $\mathbb{C}^n$.
Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of
$M_n(\mathbb{C})$, we have $G\cap \mathfrak{g}\neq\emptyset$ if and only if $G$
is a classical group and $\mathbb{C}^n$ is a minuscule representation.

Group Theory