Corollary 3

Let $u$ and $v$ be two commuting endomorphisms of E which have Jordan decompositions. a) The endomorphisms $u, v, u_s, v_s, u_n, v_n, v,$ all commute. b) The endomorphisms $u + v$ and $uv$ have Jordan decompositions with

Part a) follows from Cor. 2. To prove part b) it is enough to notice that $u_s + v_s$ and $u_s v_s$ are absolutely semi-simple (VII, p. 43, Cor.) and that $u_n + v_n$ and $u_s v_n + u_n v_s + u_n v_n$ are nilpotent (as sums of commuting nilpotent endomorphisms).