The Lie Algebras in which Every Subspace Is Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is itssubalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra isan HB Lie algebra if and only if it is isomorphic to $gdot{+}mathbb{C}id_g$, where $g$ is an abelianLie algebra. Moreover we show that the derivation algebra and the holomorph of anonabelian HB Lie algebra are complete.