Corollary 1

Assume that V is finite dimensional and that condition (AC) is satisfied. Let $k'$ be an extension of $k$. Assume that, for all $s\in S$, the endomorphism $r(s)\otimes 1$ of $V\otimes_kk'$ is triangularizable. Let $P'$ be the set of maps from S to $k'$. Then $V\otimes_kk'=\sum_{\lambda'\in P'}(V\otimes_kk')^{\lambda'}(S)$.

Let $r': S\rightarrow$ End(V$\otimes_kk'$) be the map defined by $r$. If $s_1, s_2\in S$, there exists an integer $n$ such that (ad $r(s_1)$)$^nr(s_2) = 0$, hence (ad $r'(s_1)$)$^nr'(s_2) = 0$. It now suffices to apply Th. 1.