Upper and Lower Central Series in a Pair of Lie Algebras

The Baer's theorem in the termes of the Lie algebras states that for a Lie algebra L the finiteness of \mathrm{dim}(L/Z_i(L)) implies the finiteness of\mathrm{dim}(\gamma_{i+۱}(L)). Let (N,L) denote a pair of Lie algebras, where N is an ideal of L, and d_i=d_i(L) denote the minimal number of generatorsof L/Z_i(N, L). In this paper we shall consider the pair (N, L) and show that if d_n is finite then the converse of Baer's theorem is true.In fact we shall show that if d_n and \mathrm{dim}(\gamma_{i+۱}(N, L)) are finite, where i\geq n, then N/Z_i(N, L)) is finite. In particular, we shall provide an upper bound as following,\mathrm{dim}(\frac{N}{Z_i(N, L)}) \leq ((d_n)^nd_nd_{n+۱}\ldots d_{i-۱})\mathrm{dim}(\gamma_{i+۱}(N, L))\leq (d_n)^i(\mathrm{dim}\gamma_{i+۱}(N, L)).