  Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex
matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized
by $\mathrm{U}(2,2)$ with isotropy at the origin given by
$\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of
diagonal matrices in $\mathrm{U}(2)$. We prove that the set of
$\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield
Toeplitz operators that generate commutative $C^*$-algebras on all weighted
Bergman spaces over $D$. Using tools from representation theory, we also
provide an integral formula for the spectra of these Toeplitz operators.
