Almost Lie nilpotent non-prime varieties of associative algebras

A variety of associative algebras is called Lie nilpotent if it satisfies the identity $[\cdots[[x_1,x_2],\ldots,x_n]=0$ for some positive integer $n$, where $[x,y] = xy-yx$. We study almost Lie nilpotent varieties, i.e., minimal elements in the set of all varieties that are not Lie nilpotent. We describe all almost Lie nilpotent varieties of algebras over a field of positive characteristic, both finite and infinite, in the cases when the ideals of identities of these varieties are nonprime in the class of all $T$-ideals.