  We prove that every finite dimensional unitary representation of
$\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero
$\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is no
sequence of finite-dimensional representations of $\mathrm{SL}_{4}(\mathbf{Z})$
that gives rise to an embedding of its reduced $C^*$-algebra into an
ultraproduct of matrix algebras.
