Multilinear Lie quantum operations

In is proved that if the existence condition is fulfilled the dimension of the all $n$-linear Lie quantum operations is lying between $(n-2)!$ and $(n-1)!$; moreover, the low bound is attained if the intersection of all consistent (i.e., satisfying the existence condition) subsets of a given set of “quantum” variables is nonemply. The upper bound is attained if all the subsets are consistents. The space of multilinear Lie quantum operations almost aloways is generated by symmetric operations. All exceptional cases are given. In particular, the space of general $n$-linear Lie operations is always generated by general symmetric Lie quantum operations.