I encountered a statement in Symmetries, Lie Algebras and Representations by Jurgen Fuchs& Christoph Schweigert (pp. 58) that
a lie algebra is semisimple if and only if it does not possess a non-zero solvable ideal.
I find this statement a bit confusing. Consider a non-abelian lie algebra, say $g_1$, which has a proper ideal and another simple lie algebra $g_2$. Then $g=g_1\oplus g_2$ should be non-semisimple, right? According the above statement, then $g$ must possess a non-zero solvable ideal (if not then it is semisimple). Now how to construct the non-zero solvable ideal?