Lemma 2

Let $u$ be an endomorphism of a finite dimensional vector space E over a field K. Then

Tr $u^2=$ Tr $\mathbf{S}^2(u)-$ Tr $\wedge \wedge^2(u)$.

Let $\chi_u(X) =\prod_{i=1}^n(X-\alpha_i)$ be a decomposition of the characteristic polynomial of $u$ into linear factors in a suitable extension of K. We have Tr $u^2=\sum_i\alpha^2_i$, Tr $\wedge \wedge^2(u) =\sum_{i<j}\alpha_i\alpha_j$, Tr $\mathbf{S}^2(u) =\sum_{i\leq j}\alpha_i\alpha_j$ (cf. Algebra, Chap. VII, §5, no. 5, Cor. 3), hence the result.