Suppose we have two unary operations $f,g$, related by the law
$$\forall a,\quad g(f(a))=f(f(g(g(a)))),$$
that is, $gf=ffgg$. We might then wonder whether any expression involving these operations can be written with all $f$'s on the left and all $g$'s on the right. Let's try an example:
$$gff$$ $$=(ffgg)f$$ $$=ffg(gf)$$ $$=ffg(ffgg)$$ $$=ff(gff)gg$$ $$=ff\big(ff(gff)gg\big)gg$$ $$=ff\Big(ff\big(ff(gff)gg\big)gg\Big)gg$$
Evidently, applying that law can never get rid of the $gff$ in the middle.
On the other hand, if the law is $gf=f^mg^n$ where either $m\leq1$ or $n\leq1$, then applying it to any expression does eventually end with all $f$'s on the left and all $g$'s on the right.
Suppose we have two (commutative, associative) binary operations $\oplus,\odot$, related by the law
$$a\odot(b\oplus c)=(a\odot a)\oplus(b\odot c).$$
As before, and in analogy with the ordinary distributive law, we might wonder whether any expression involving these operations can be written in the standard form (i.e. with multiplication inside and addition outside of parentheses). As before, the answer is negative; there's another self-replicating expression:
$$(a\oplus b)\odot(a\oplus b)$$ $$=\big((a\oplus b)\odot(a\oplus b)\big)\oplus(a\odot b)$$
So what about other modifications of the distributive law,
$$a\odot(b\oplus c)=P(a,b,c)$$
where $P$ is a polynomial in the standard form? What conditions on $P$ ensure that any expression can also be written in the standard form?