${\mathrm{S}\mathrm{L}}_4(\mathbf{Z})$ is not purely matricial field

We prove that every finite dimensional unitary representation of ${\mathrm{S}\mathrm{L}}_4(\mathbf{Z})$ contains a non-zero ${\mathrm{S}\mathrm{L}}_2(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of ${\mathrm{S}\mathrm{L}}_4(\mathbf{Z})$ that gives rise to an embedding of its reduced $C^{\ast }$-algebra into an ultraproduct of matrix algebras.