Identities in differential perm algebras

Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic $0$, i.e. an associative algebra satisfying $(ab)c=(ba)c$ equipped with a derivation $d$. We investigate polynomial identities in the algebras obtained from $d$ by the derived operations $$a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab',$$ where $a'=d(a)$. Our first result shows that any nontrivial differential polynomial identity (not supported by the right annihilator forced by the perm law) implies a purely differential consequence of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $\blacklozenge$ and under $\bullet$, giving explicit generating sets and computing the multilinear dimensions of their homogeneous components. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.