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We prove that every finite dimensional unitary representation of$\mathrm{SL}_{4}(\mathbf{Z})$ contains an$\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is nosequence of finite-dimensional representations of $\mathrm{SL}_{4}(\mathbf{Z})$that gives rise to an embedding of its reduced $C^*$-algebra into anultraproduct of matrix algebras.

$\mathrm{SL}_{4}(\mathbf{Z})$ is not pure matrix field