Noncommutative Abel-like identities

We generalize the Abel–Hurwitz identities to an almost entirely noncommutative setting. Namely, let $V$ be a finite set of size $n$, and let $\mathbb{L}$ be any noncommutative ring. For each $s\in V$, let $x_s\in \mathbb{L}$. Set $x\left(S\right):={\sum }_{s\in S}{x}_s$ for any $S\subseteq V$. Let $X$ and $Y$ be two elements of $\mathbb{L}$ such that $X+Y$ lies in the center of $\mathbb{L}$. Then, we show that% \begin{align*} & \sum_{S\subseteq V}\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert }\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\sum_{\substack{i_{1},i_{2},\ldots,i_{k}\in V$\text{ distinct}$% }}\left( X+Y\right) ^{n-k}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}};\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\left( X+Y\right) ^{n};\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert -1}\left( Y-x\left( V\right) \right) =\left( X+Y-x\left( V\right) \right) \left( X+Y\right) ^{n-1}. \end{align*} (Negative powers are understood to be cancelled by other factors.)