pairs of finite dimensional nilpotent and filiform lie algebras

Let (n,l) be a pair of finite dimensional nilpotent lie algebras. if n admits a complement k in l such that dim n = n and dim k = m, then dim m(n,l) = 1/2n(n + 2m- 1) t(n,l), where m(n,l) is the schur multiplier of the pair (n,l) and t(n,l) is a nonnegative integer. in this paper, we characterize the pair (n,l) for t(n,l)=0, 1, 2, … , 23, where n is a finite dimensional filiform lie algebra and n,k are ideals of l such that l = n ⊕ k. moreover, we classify the pair (n,l) for s′ (n,l) = 3, where s′ (n,l) = 1/2 (n- 1)(n -2) + 1 + (n- 1)m – dim m(n,l), l is a finite dimensional nilpotent lie algebra and n is a nonabelian ideal of l.