Let $X$ be a Banach space.

Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $X_1,...,X_n$. Is there a formula for a projection onto $X_1+...+X_n$ (it terms of $P_1,...,P_n$) (of course, under certain conditions)?

Remark. I have some results on this question. However, the question seems to be a very basic question in the theory of complemented subspaces, so I have doubts about originality of my results. The only references I have found are results by Alan LaVergne (paper "Remark on sums of complemented subspaces"), Lars Svensson (paper "Sums of complemented subspaces in locally convex spaces"), Manuel Gonzalez (paper "On essentially incomparable Banach spaces", Lemma 1), and $S\ddot{u}leyman$ $\ddot{O}nal$, Murat Yurdakul (paper "On sums of complemented subspaces").

My results on the question are presented in my recent paper "When is the sum of complemented subspaces complemented?" https://arxiv.org/abs/1606.08048.

I would be very grateful for your comments.