For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies
$$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left((I_{3} + \mathcal{M})G(x)\right) \wedge G^{\prime}(x)$$
and that moreover
$$\|G^{\prime}(0)\| = 1, \quad \left((I_{3} + \mathcal{M})G(0)\right) \cdot G^{\prime}(0) = 0$$
We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$.
Show that for all $x \in \mathbb{R}$, $\left(I_{3} + \mathcal{M}\right)G(x) - x G^{\prime}(x) = 2 G^{\prime}(x) \wedge G^{\prime\prime}(x)$.