Recognition of identities in quotient algebras of universal enveloping algebras

For special type of (associative) polynomials $f$ and simple algebras $L$ the problem of recognition of identity $f$ in quotient algebra $U_L/J$ of universal enveloping algebra $U_L$ by arbitrary ideal $J$, where $J$ is given by its generators, is solved. The central point of the solution is the
Theorem. Let $l_1,\ldots,l_p$ be Lie (associative) polynomials with non-intersecting sets of variables which are not identities in $L$, $f=\prod\limits_{i=1}^{p}l_{i}(x_{i_{1}},\ldots,x_{i_{n_{i}}})$, then the verbal ideal $T_f(U_L)$ generated by polynomial $f$ in $U_L$ is equal to $U_L{}^p$.
In particular, $U_L/T_f(U_L)$ is a nilpotent algebra of degree $p$.