Let $A,B,C$ be subalgebras of some algebra $X$. I've managed to show, that $A\cap C+B\cap C\subseteq (A+B)\cap C$. If $x\in A\cap C+B\cap C$, then $x=a+b$, where $a\in A\cap C, b\in B\cap C$. Since $a,b\in C$ then $a+b=x\in C$. Since $a\in A, b\in B$ then $a+b=x\in (A+B)$, so $x\in (A+B)\cap C$.
Does the reverse inclusion holds? I could neither show it nor disprove it.
Here $A+B$ denotes subalgebra $\{a+b: a\in A, b\in B\}$.