Closed form for the exponential of a Lie algebra 3x3 matrix? 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 $u_{jk}$?
 A: 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.
A: Call complex NxN matrices whose rows and columns sum to zero "magic". These matrices are closed under addition and multiplication and so are closed under commutation and so form a Lie algebra (physics definitions here, bring in a factor of i if you like). There is a "Householder" transformation that shows that the magic NxN complex matrices are equivalent to the (N-1)x(N-1) complex matrices so an explicit exponentiation of N-1 square matrices defines an explicit exponentiation for NxN magic matrices. See http://brannenworks.com/Gravity/jmapumm.pdf for details on the Householder transformation that performs this.
Exponentiating the 2x2 complex matrices is easy. Write them using a basis of the Pauli spin matrices plus 2x2 identity. The identity commutes with the Pauli spin matrices and so is easy to exponentiate. The Pauli matrices exponentiate easily using the fact that $u_x\sigma_x + u_y\sigma_y + u_z\sigma_z)^2 = (u_x^2+u_y^2+u_z^2)\;1$ where 1 is the 2x2 identity matrix and $u_j$ are complex numbers.
