Problem with the definition of simple Lie algebra

Problem with the definition of simple Lie algebra

I have the following definitions:
Simple Lie algebra: A Lie algebra $\mathfrak{g}$ is simple if it ha no
non-trivial (i.e. $0$ or $\mathfrak{g}$ itself) ideals.
Semisimple Lie algebra: A Lie algebra $\mathfrak{g}$ is semisimple if it
has no non-trivial (i.e. $0$) solvable ideals.
And the following result:
Claim: A Lie algebra $\mathfrak{g}$ is semisimple if, and only if it is
(isomorphic to) the direct sum of simple Lie algebras.
Now I have a problem with this. Indeed let $\mathfrak{g} =
\mathbb{C}\oplus \mathfrak{h}$ where $\mathfrak{h}$ is any simple Lie
algebra. Then $\mathbb{C}$ is abelian (and thus solvable), and an ideal of
$\mathfrak{g}$. At the same time it satisfies the definition of a simple
Lie algebra, and so by the claim $\mathfrak{g}$ should be semisimple.
Obviously there's something wrong. I think the problem is in the
definition of "simple Lie algebra". Can anyone confirm that, or explain
where I'm wrong/what I'm not seeing?