Let $A=(a_{ij})_{i,j=1}^{4}\in M_4(\mathbb R)$. Apply inverse of the isomorphis $\chi:\mathbb H\to M_4(\mathbb R)$ where $\chi(a)=A$ and $$\chi (a_{0}+a_{1}i+a_{2}j+a_{3}k)=\begin{bmatrix} a_{0} & -a_{1} & a_{3} & -a_{2} \\ a_{1} & a_{0} & -a_{2} & -a_{3} \\ -a_{3} & a_{2} & a_{0} & -a_{1} \\ a_{2} & a_{3} & a_{1} & a_{0} \end{bmatrix}.$$ Since every real matrix with size $4\times 4$ is represented by a quaternion, I want to find $a$ as $a=a_0+a_1 i+a_2 j+a_3 k$.
I don't know how i use inverse of $\chi$.
Hint:
$\chi$ is not an isomorphism from $ \mathbb H$ and $M_4(\mathbb R)$, but from $ \mathbb H$ and the subspace $V$ of $M_4(\mathbb R)$ spanned by the matrices: $$ E= \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} \qquad I= \begin{bmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{bmatrix} $$ $$ J= \begin{bmatrix} 0&0&0&-1\\ 0&0&-1&0\\ 0&1&0&0\\ 1&0&0&0 \end{bmatrix} \qquad K= \begin{bmatrix} 0&0&1&0\\ 0&0&0&-1\\ -1&0&0&0\\ 0&1&0&0 \end{bmatrix} $$ With the correspondence $$ 1 \to E\quad i \to I \quad j \to J \quad k \to K $$ Note that $V$ is a space of dimension $4$ over $\mathbb R$ and $\chi$ is invertible from $V$ to $\mathbb H$ in a an obvious way.