Multilinear quantum Lie operations

It is shown that the dimension of the multilinear quantum Lie operations space is either equal to zero or included between $(n-2)!$ and $(n-1)!.$ The lower bound is achieved if the intersection of all conforming subsets is nonempty, while the upper bound does if all subsets are conforming. We show that almost always the quantum Lie operations space is generated by symmetric ones. In particular, the space of all general $n$-linear quantum Lie operations does. All possible exceptions are described.