determining whether Lie algebra is enlarged by new generator

Question

Is there a simple criterion to determine whether, given a set of generators $A_1,\ldots,A_n$ of a Lie subalgebra $\mathfrak{h}\subset\mathfrak{g}$, adding a new element $B\in\mathfrak{g}$ will enlarge the subalgebra? Clearly if $B$ is in the linear span of $A_1,\ldots,A_n$, the subalgebra is not enlarged, but if this is not the case, is there anything to do other than actually list out all of the nested commutators of the $A_i$ and check that $B$ is linearly independent of them?