Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$?

Actually I need to know is there any algebra $A$ having signature $\mathcal{L}$ such that for any $n$-ary $f\in \mathcal{L}$ and any $m$-ary $g\in\mathcal{L}$ we have $$ f(g(x_{11}, \ldots, x_{1m}), \ldots, g(x_{n1}, \ldots,x_{nm}))=g(f(x_{11}, \ldots,x_{n1}), \ldots, f(x_{1m},\ldots, x_{nm})) $$ Also I need to have all operations idempotent. As my motivation, I should say that for any variety of such algebras, we have the following interesting property: Let $F=F_V(x_1,\ldots,x_n)$ be a relative free algebra in $V$. Then the solution set of any system of equations in $F$ is a subalgebra of $F^k$ where $k$ is the number of indeterminate.

P.S. Instead of commutativity, consider the next two weaker conditions:

$(a+b)+(c+d)=(a+c)+(b+d)$.

$(ab)(cd)=(ac)(bd)$. \

P.S.2. It is good to use the phrase Medial instead of Commutative. So my question becomes: Is there a known example of an algebra $(A, +, \cdot)$ with two binary medial and idempotent operations satisfying the identity $(a+b)(c+d)=ac+bd$?