Literature and history for: lifting matrix units modulo various kinds of ideal

This is not so much a mathematics question as a cross between a "history of mathematics" question and a reference request.

My PhD student has been working on some problems concerning subhomogeneous Banach algebras, and at one point this requires lifting matrix units from $A/J$ to $A$, where $A$ is a Banach algebra and $J$ a closed ideal with certain properties. In our setting, the ideal $J$ is not contained in the Jacobson radical (indeed, $A$ has trivial Jacobson radical), but $J$ does have some special features that allow one to adapt various proofs from the “$J$ is radical” setting.

The arguments that my student is adapting come from Rowen’s book on Ring Theory, but I wondered where one could find the initial literature about such lifting theorems. The proof in Rowen’s book is fairly direct but it has the effect of making the calculations seem "pulled out of a hat", rather than the consequence of some bigger conceptual picture; and we would like to have some idea of the bigger picture, both for general acknowledgement of the literature and for possible inspiration in the Banach-algebraic setting.

Here are some specific questions. My hope is that some members of the MO community who specialise in topics adjacent to noncommutative ring theory will have seen appropriate articles or textbook expositions.
We assume characteristic zero throughout (and all algebras are associative).

Q1. If $A$ is a finite-dimensional algebra and $J$ is its Jacobson radical then the quotient map $q: Ato A/J$ has a right inverse homomorphism $sigma: A/J to A$ (Wedderburn decomposition). In particular, matrix units in $A/J$ will lift to matrix units in $A$. Did Wedderburn’s original paper make this observation, either as a corollary or as an ingredient of the decomposition theorem? Are there other pre-1950 sources which explicitly link the lifting of matrix units modulo the radical to Wedderburn decomposition?

Q2. If $A$ and $J$ are as in Q1 and one knows in advance that $A/J$ is a sum of matrix algebras, then one can use an iterative argument based on vanishing Hochschild cohomology groups $H^2(A/J, underline{quad})$ to construct the splitting map $sigma$. Has this approach been extended to provide lifting of matrix units in settings where $A$ is infinite-dimensional and $J$ is no longer a nil ideal?

Q3. Following on from Q2, are there some standard references for lifting results in settings where $J$ is not (contained in) the Jacobson radical?

Q4. It seems much easier to find literature on lifting orthogonal idempotents modulo ideals that satisfy various properties. Am I missing some "obvious" reason why the problem of lifting matrix units reduces to this problem? (In Rowen’s book, given certain technical conditions on $J$, the first step in lifting matrix units modulo $J$ is to lift the diagonal entries to orthogonal idempotents; is this a matter of technical convenience, or done to follow some deeper principle?)