  We construct a quadratic basis of generators of matrix-extended
$\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation.
This makes it possible to conjecture a closed-form formula for the operator
product expansions defining the algebra. We study truncations of the algebra.
An explicit calculation at low levels shows that these are parametrized in a
way consistent with the gluing description of the algebra. It is perhaps
surprising that in spite of the fact that the algebras are rather complicated
and non-linear, the structure of their truncations follows very simple gluing
rules.
