I'm checking right now a book, in which the authors say, that they identify the dual spaces of the lie algeba $u(n) = \{A \in \mathbb{C}^{n \times n} | \overline{A} = -A^T \}$ again with $u(n)$ by using the Killingform.
But this shoudn't be possible? As $u(n)$ is not semisimple, the Killingform is degenerate, therefore it shouldn't be even an isomorphism...
So is there some other possibility to identify the dual of $u(n)$ with $u(n)$?