For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that for all $t \in \mathbb{R}$ and $X, Y$ vectors of $\mathbb{R}^{3}$, we have $F(t)X \cdot Y = X \cdot F(-t)Y$. Deduce that $F(t)(X \wedge Y) = (F(t)X) \wedge (F(t)Y)$.