Levi decomposition of a quiver automorphism group

For a finite-dimensional quiver representation $X$ over an algebraically closed field $k$, fix a Krull-Schmidt decomposition $X={⨁}_a{M}_a^{\oplus m_a}$ with pairwise nonisomorphic indecomposable modules. Acting by scalar matrices on the multiplicity spaces embeds $L={\prod }_a{\mathrm{GL}}_{m_a}(k)$ into ${\mathrm{Aut}}_Q(X)$. It splits reduction modulo the Jacobson radical $J=J({\mathrm{End}}_Q(X))$, givingThe subgroup $1+J$ is closed, normal and unipotent, because $J$ is a nilpotent ideal. This explicit section works in every characteristic. The algebraically closed field assumption ensures that the residue division rings of the indecomposable endomorphism rings are $k$; without it this particular product of split general linear groups need not be the quotient.