on nilpotency of outer pointwise inner actor of the lie algebra crossed modules

Let l be a lie algebra crossed module and actpi(l) be a pointwise inner actor of l. in this paper, we introduce lower and upper central series of l and show that if actpi( l / zj (l) )/innact( l / zj (l) ) is the nilpotent of class k wherein zj (l) denotes the nth term of the upper central series of l , then actpi(l)/innact(l) is the nilpotent of the maximum class j + k. moreover, if dim(li /(li ∩ zj (l))) ≤ (1, 1), then actpi(l)/innact(l) is the nilpotent of the maximum class i + j − 1.