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 $$a\prec b=a{b}^{\mathrm{\prime }},a\succ b=a^{\mathrm{\prime }}b,a\mathrm{⧫}b=a{b}^{\mathrm{\prime }}+b{a}^{\mathrm{\prime }},a\bullet b=a^{\mathrm{\prime }}b+a{b}^{\mathrm{\prime }},a\mathrm{◊}b=a{b}^{\mathrm{\prime }}-b{a}^{\mathrm{\prime }},a\circ b=a^{\mathrm{\prime }}b-a{b}^{\mathrm{\prime }},$$ where $a^{\mathrm{\prime }}=d(a)$. 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 $a_1^{\mathrm{\prime }}{a}_2^{\mathrm{\prime }}\cdots a_m^{\mathrm{\prime }}=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $\mathrm{⧫}$ 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.