If $X^\dagger$ denotes transposed conjugate, since $tr(X^\dagger Y)$ is the Hermitian inner product on $\mathfrak{gl}(4,\mathbb C)$, the level sets are two planes perpendicular to $G$ with the same distance to 0, intersected with $SU(4)$.

If $X^\dagger$ denotes transposed conjugate, since $tr(X^\dagger Y)$ is the Hermitian inner product on $\mathfrak{gl}(4,\mathbb C)$, the level sets are codimesion 1 circular cylinders (circles in the complex plane $\mathbb C.G$ times the orthogonal complement of $G$) intersected with $SU(4)$.