\left(\begin{array}{cc} S & 0 \\ 0 & S \end{array} \right)$$ with $$S = \left(\begin{array}{cc} 0 & -I \\ {\mathrm{id}}_3 & 0 \end{array} \right)$$ where $I$ is the following “inertia matrix”: $$I = \left(\begin{array}{ccc} Y_2^2+Y_3^2 & -Y_1Y_2 & -Y_1Y_3 \\ -Y_1Y_2 & Y_1^2+Y_3^2 & -Y_2Y_3 \\ -Y_1 Y_3 & -Y_2 Y_3 & Y_1^2+Y_2^2 \end{array} \right)$$ with $Y_i = X_i-x_i$. ${\mathrm{Sp}}(S) \subset \imath \mathbb R$, indeed let $\lambda$ be an eigenvalue of $S$: $Se = \lambda e$. $e = (\mathring e,\hat e)$ with $\mathring e, \hat e \in \mathbb C^3$. The eigenequation is equivalent to $\left\{ \begin{array}{rcl} - I \hat e & = & \lambda \mathring e \\ \mathring e & = & \lambda \hat e \end{array} \right.$, and then $-I \hat e = \lambda^2 \hat e$. $\lambda^2 \in {\mathrm{Sp}}(-I)$, but $I \geq 0$ and then $\lambda^2 \leq 0 \iff \lambda \in \imath \mathbb R$. $S$ being real, its spectrum contains three couples of conjugated purely imaginary eigenvalues.\
The 12 local Lyapunov values of the dynamical system in the neighbourhood of $0$ being purely imaginary, the fixed point is simply stable, the evolution of $\delta x$ is then quasi-periodic with 3 fundamental frequencies (the eigenvalues of $\sqrt I$) and the phase trajectory 