Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.

For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical projection $V^{\otimes n}\to \mathrm{Sym}^n V$.

Let $m\in\mathbb N$ be such that $m\geq 2$.

Let $\rho$ be the canonical map $K^{m-1}\left(A\right)\otimes B\to A^{\otimes \left(m-1\right)}\otimes B$.

Let $\phi$ be the canonical map $A^{\otimes \left(m-1\right)}\otimes B \to A^{\otimes \left(m-1\right)}\otimes A = A^{\otimes m}$.

Let $\psi$ be the canonical map $A^{\otimes \left(m-2\right)} \otimes K^2\left(B\right) \to A^{\otimes \left(m-2\right)} \otimes B^{\otimes 2} \to A^{\otimes \left(m-2\right)} \otimes A^{\otimes 2} = A^{\otimes m}$.

Under what conditions do we have $K^m\left(A\right) \cap \phi\left(A^{\otimes \left(m-1\right)}\otimes B\right) = \phi\left(\rho\left(K^{m-1}\left(A\right)\otimes B\right)\right) + \psi\left(A^{\otimes \left(m-2\right)} \otimes K^2\left(B\right) \right)$ ?

(Note that when $k$ is a field, or under appropriately strong flatness conditions, we can think about the maps $\rho$, $\phi$ and $\psi$ as just being the obvious embeddings.)

I would like to see flatness or projectivity conditions, but at the moment I am not even sure that it holds when $k$ is a field. If you can find a condition better than "both $B$ and $A/B$ are projective", you have improved a result from my diploma thesis about relative Poincaré-Birkhoff-Witt theorems. But I think the question is interesting independently of Lie algebras, as it helps understanding $K^m$.