Matrices of the form: $\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$ where $x,y,z,w$ may be assumed to be real, form a Lie algebra. That is, such matrices are closed under addition and multiplication, and include inverses, zero, and unity. Each row or column sums to zero. Can the exponentiation of this matrix be put into closed form?
That is, can we solve: $\begin{pmatrix} u_{11}&u_{12}&u_{13}\\ u_{21}&u_{22}&u_{23}\\ u_{31}&u_{32}&u_{33}\end{pmatrix} = \exp \begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$ in closed form for all the $u_{jk}$?