Nilpotent implies center is ideal-large

Statement
If $$L$$ is a nilpotent Lie ring, the center of $$L$$ is an ideal-large Lie subring: its intersection with any nonzero ideal of $$L$$ is nonzero.

Related facts

 * Nilpotent implies center is normality-large is the analogous statement for groups. The proof is analogous.