Remark

(1) If $(M_i)_{0 \leq i \leq r}$ is replaced by another Jordan-Hölder series of $M$, the sequence $(\lambda_1, \ldots, \lambda_r)$ is replaced by a sequence of the form $(\lambda_{\pi(1)}, \ldots, \lambda_{\pi(r)})$, where $\pi$ is a permutation of $\{1, \ldots, r\}$, as follows from the Jordan-Hölder Theorem.

(2) Let $(e_1, \ldots, e_r)$ be a basis of $M$ such that $e_i \in M_{i-1}, e_i \notin M_i$ ($1 \leq i \leq r$). If $x \in g$ the endomorphism of $M$ corresponding to $x$ is represented with respect to this basis by a triangular matrix whose diagonal coefficients are