To be precise, fix $n$, fix a field $k$.

What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean all the products of matrices in this subspace are commutative.

(I feel like this can be formulated in terms of Lie algebras, but I don't find a good one. And I think the down-to-earth formulation might make it more accessible.)