Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

Question

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where:

$A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z \otimes \sigma_z)$ for some real numbers $J^k$ and

$B_1 = i\sigma_z \otimes I$ and $B_2 = I \otimes \sigma_z$.

NB: In the above definitions, $\sigma_x$, $\sigma_y$, and $\sigma_z$ are the Pauli matrices, $P\otimes Q$ means the Kronecker product of matrices, and the constants $J^x$, $J^y$, and $J^z$ are fixed but arbitrary.

Answer 1

The answer depends on the values of the constants $J^x$, $J^y$, and $J^z$. Here is what direct computation yields:

If $J^x=J^y=J^z=0$, so that $A=0$, then $B_1$ and $B_2$ span a $2$-dimensional abelian subalgebra.

If $J^x=J^y=0$, but $J^z\not=0$, then $A$, $B_1$, and $B_2$ span a $3$-dimensional abelian subalgebra (i.e., a maximal torus in ${\frak{su}}(4)$.

If $J^x = \pm J^y \not = 0$ and $J^z=0$, then they generate a $4$-dimensional subalgebra isomorphic to $\mathbb{R}\oplus {\frak{su}}(2)$.

If $J^x = \pm J^y \not = 0$ and $J^z\not=0$, then they generate a $5$-dimensional subalgebra isomorphic to $\mathbb{R}\oplus\mathbb{R}\oplus {\frak{su}}(2)$.

If $J^x \not= \pm J^y $ and $J^z=0$, then they generate a $6$-dimensional subalgebra isomorphic to ${\frak{su}}(2)\oplus {\frak{su}}(2)$.

If $J^x \not= \pm J^y $ and $J^z\not=0$, then they generate a $7$-dimensional subalgebra isomorphic to $\mathbb{R}\oplus {\frak{su}}(2)\oplus {\frak{su}}(2)$.

All of these follow by direct computation with matrices, which is made easier, when you write them out, by interchanging the second and fourth columns and rows, because then everything is conjugated into the Lie subalgebra of $S\bigl(SU(2)\times SU(2)\bigr)$.