  We present an explicit matrix algebra regularization of the algebra of
volume-preserving diffeomorphisms of the $n$-torus. That is, we approximate the
corresponding classical Nambu brackets using
$\mathfrak{sl}(N^{\lceil\tfrac{n}{2}\rceil},\mathbb{C})$-matrices equipped with
the finite bracket given by the completely anti-symmetrized matrix product,
such that the classical brackets are retrieved in the $N\rightarrow \infty$
limit. We then apply this approximation to the super $4$-brane in $9$
dimensions and give a regularized action in analogy with the matrix
regularization of the supermembrane. This action exhibits a reduced gauge
symmetry that we discuss from the viewpoint of $L_\infty$-algebras in a slight
generalization to the construction of Lie $2$-algebras from Bagger-Lambert
$3$-algebras.
