Remark 2

A necessary and sufficient condition for $u$ to be triangularisable is that $u$ have a Jordan decomposition with $u_s$ diagonalisable. Then there exists a basis of E with respect to which the matrix of $u$ is lower triangular, and that of $u_s$ is diagonal, with the same diagonal as the matrix of $u$ (cf. Lemma 4 and Prop. 19 below). Note however that if the matrix of $u$ with respect to some basis is triangular, it does not in general follow that the matrix of $u_s$ with respect to the same basis is diagonal.