We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl}
\forall i & S_i \circ S_i & = \operatorname{Id}_E \\
\forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0
\end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned}
\forall i & A_i^2 & = I_n \\
\forall i \neq j & A_i A_j + A_j A_i & = 0
\end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the length $p$ of an H-system of endomorphisms of $E$ is bounded above by $n^2$.
We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that
$$\left\{ \begin{array}{lrl}
\forall i & S_i \circ S_i & = \operatorname{Id}_E \\
\forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0
\end{array} \right.$$
Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that
$$\left\{ \begin{aligned}
\forall i & A_i^2 & = I_n \\
\forall i \neq j & A_i A_j + A_j A_i & = 0
\end{aligned} \right.$$
In both cases, $p$ is called the length of the H-system.\\
Show that the length $p$ of an H-system of endomorphisms of $E$ is bounded above by $n^2$.