Proposition 7

Let $(\mathfrak{g}_i)_{i\in I}$ be a finite family of Lie algebras, and let $\mathfrak{g}=$ $\prod_{i\in I}\mathfrak{g}_i$.

(i) An element $(x_i)_{i\in I}$ of $\mathfrak{g}$ is regular in $\mathfrak{g}$ if and only if, for all $i\in I,x_i$ is regular in $\mathfrak{g}_i$.

(ii) rk($\mathfrak{g}$) $=\sum_{i\in I}$ rk($\mathfrak{g}_i$).

Indeed, for any $x= (x_i)_{i\in I}\in \mathfrak{g}$, the characteristic polynomial of ad$_{\mathfrak{g}}x$ is the product of the characteristic polynomials of the ad$_{\mathfrak{g}_i}x_i$.