We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $N = p(n)$ and $(a_1, \ldots, a_N)$ be an H-system of matrices of size $n$, that is, such that $$\forall i, a_i^2 = I_n \quad \text{and} \quad \forall i \neq j, a_i a_j + a_j a_i = 0$$ By considering the following matrices of $\mathcal{M}_{2n}(\mathbb{C})$ written in block form $$A_j = \left( \begin{array}{cc}
a_j & 0 \\
0 & -a_j
\end{array} \right) (j \in \llbracket 1, N \rrbracket), \quad A_{N+1} = \left( \begin{array}{cc}
0 & I_n \\
I_n & 0
\end{array} \right), \quad A_{N+2} = \left( \begin{array}{cc}
0 & \mathrm{i} I_n \\
-\mathrm{i} I_n & 0
\end{array} \right)$$ show that $p(2n) \geqslant N + 2$.
We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$.\\
Let $N = p(n)$ and $(a_1, \ldots, a_N)$ be an H-system of matrices of size $n$, that is, such that
$$\forall i, a_i^2 = I_n \quad \text{and} \quad \forall i \neq j, a_i a_j + a_j a_i = 0$$
By considering the following matrices of $\mathcal{M}_{2n}(\mathbb{C})$ written in block form
$$A_j = \left( \begin{array}{cc}
a_j & 0 \\
0 & -a_j
\end{array} \right) (j \in \llbracket 1, N \rrbracket), \quad A_{N+1} = \left( \begin{array}{cc}
0 & I_n \\
I_n & 0
\end{array} \right), \quad A_{N+2} = \left( \begin{array}{cc}
0 & \mathrm{i} I_n \\
-\mathrm{i} I_n & 0
\end{array} \right)$$
show that $p(2n) \geqslant N + 2$.