Proposition 8

Let $E$ be a finite dimensional vector space over a commutative field $K$, and let $u$ be an endomorphism of $E$. Then the following conditions are equivalent:

(i) the eigenvalues of $u$ (in some algebraically closed extension of $K$) belong to $K$;

(ii) there exists a basis of $E$ with respect to which the matrix of $u$ is lower (resp. upper) triangular;

(iii) there exists a basis of $E$ with respect to which the matrix of $u$ is a diagonal block of Jordan matrices.

We have (i) $\Rightarrow$ (iii) by Prop. 7 and the above remarks, and the assertions (iii) $\Rightarrow$ (ii) and (ii) $\Rightarrow$ (i) are trivial.