# Is every nilpotent Lie algebra the nilradical of some Lie algebra?

If we take $F=\mathbb{C}$ and assume the Lie algebras involved are finite dimensional, is it true that if $\mathfrak{g}$ is nilpotent then it is the nilradical of some other Lie algebra $\mathfrak{h}$?

• Every nilpotent algebra is its own nilradical. – Mariano Suárez-Álvarez Aug 22 '17 at 23:58
• If you do not like that, then consider the direct sum of your nilpotent lie algebra and any semisimple one. – Mariano Suárez-Álvarez Aug 23 '17 at 0:00