Let (N, L) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L and ℳ(N, L) is the Schur multiplier of the pair (N, L). In 2011, the second author and colleagues proved that if N admits a complement K say, in L with dim N = n and dim K = m, then , where t(N, L) is a non-negative integer. In the present article, we characterize the pairs (N, L) for which t(N, L) = 0, 1, 2, 3, and 4. Also, in a special case when L is a filiform Lie algebra, we determine all pairs (N, L), when t(N, L) = 0, 1,…, 10.