Example 3

Let V be a vector space of finite dimension $n$, and $\mathfrak{g}=\mathfrak{g}\mathfrak{l}(V)$. Let $\overline{k}$ be an algebraic closure of $k$. Let $x\in \mathfrak{g}$, and let $\lambda_1, . . . , \lambda_n$ be the roots in $\overline{k}$ of the characteristic polynomial of $x$ (each root being written a number of times equal to its multiplicity). The canonical isomorphism from $V^*\otimes V$ to $\mathfrak{g}$ is compatible with the $\mathfrak{g}$-module structures of these two spaces, in other words it takes $1\otimes x-^tx\otimes 1$ to ad $x$ (Chap. I, §3, no. 3, Prop. 4). In view of §1, Prop. 4 (i), it follows that the roots of the characteristic polynomial of ad $x$ are the $\lambda_i-\lambda_j$ for $1\leq i\leq n,1\leq j\leq n$ (each root being written a number of times equal to its multiplicity). Thus, the rank of $\mathfrak{g}$ is $n$, and $x$ is regular if and only if each $\lambda_i$ is a simple root of the characteristic polynomial of $x$.