  I. Penkov and V. Serganova have recently introduced, for any non-degenerate
pairing $W\otimes V\to\mathbb C$ of vector spaces, the Lie algebra
$\mathfrak{gl}^M=\mathfrak{gl}^M(V,W)$ consisting of endomorphisms of $V$ whose
duals preserve $W\subseteq V^*$. In their work, the category
$\mathbb{T}_{\mathfrak{gl}^M}$ of $\mathfrak{gl}^M$-modules which are finite
length subquotients of the tensor algebra $T(W\otimes V)$ is singled out and
studied. In this note we solve three problems posed by these authors concerning
the categories $\mathbb{T}_{\mathfrak{gl}^M}$. Denoting by
$\mathbb{T}_{V\otimes W}$ the category with the same objects as
$\mathbb{T}_{\mathfrak{gl}^M}$ but regarded as $V\otimes W$-modules, we first
show that when $W$ and $V$ are paired by dual bases, the functor
$\mathbb{T}_{\mathfrak{gl}^M}\to \mathbb{T}_{V\otimes W}$ taking a module to
its largest weight submodule with respect to a sufficiently nice Cartan
subalgebra of $V\otimes W$ is a tensor equivalence. Secondly, we prove that
when $W$ and $V$ are countable-dimensional, the objects of
$\mathbb{T}_{\mathrm{End}(V)}$ have finite length as $\mathfrak{gl}^M$-modules.
Finally, under the same hypotheses, we compute the socle filtration of a simple
object in $\mathbb{T}_{\mathrm{End}(V)}$ as a $\mathfrak{gl}^M$-module.
