Subalgebras of matrices property

Question

Let $g$ be a Lie subalgebra of $gl_n(\mathbb{C})$ which has the propety that if $a\in g$ then also $a^\dagger\in g$ (where $a^\dagger$ is conjugate transpose). I want to show that if $a$ is an ideal of $g$ then there is an ideal $b$ of $g$ with $g=a\oplus b$. I guess that given $a$ I should define $b=a^\dagger=\{x^\dagger \ | \ x\in a\}$. I can check that this is an ideal, but not that it has the above properties. Am I taking the wrong $b$?