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 $u_{jk}$?
First of all, to be a Lie algebra it has to be closed under commutator, not matrix multiplication. This one is not closed under matrix multiplication (if you require the x,y,z,w real), although it is closed under commutator. As for the matrix exponential, that can be computed in closed form, e.g. by Maple using the MatrixExponential command. The result, however, is very complicated, too big to be copied here.