= Solution
For a finite-dimensional complex <Lie algebra> $\mathfrak g$, the following three conditions are equivalent descriptions of a <semisimple Lie algebra>.
First, its <solvable radical> is zero. An <ideal of a Lie algebra> is a linear subspace $I$ satisfying $[\mathfrak g,I]\subseteq I$. A <solvable Lie algebra> is one whose <derived series>, defined by $I^{(0)}=I$ and $I^{(r+1)}=[I^{(r)},I^{(r)}]$, eventually becomes zero. The <solvable radical> is the largest solvable <ideal of a Lie algebra>; its vanishing is equivalent to the absence of nonzero solvable ideals.
Second, the <Killing form>
$$
\kappa(u,v)=\operatorname{tr}(\operatorname{ad}u\operatorname{ad}v)
$$
is a <nondegenerate bilinear form>: if $\kappa(u,v)=0$ for every $v$, then $u=0$. Here $\operatorname{ad}u(v)=[u,v]$ is the <Adjoint representation>, and $\operatorname{tr}$ is the <trace>. This equivalence is the <Cartan criterion for semisimplicity>.
Third, $\mathfrak g$ is a <direct sum of Lie algebras> that are nonabelian <simple Lie algebras>. A <simple Lie algebra> is nonabelian and has no <ideals of a Lie algebra> other than zero and itself. Thus the three equivalent criteria are
$$
\boxed{\operatorname{rad}\mathfrak g=0\quad\Longleftrightarrow\quad\kappa\text{ nondegenerate}\quad\Longleftrightarrow\quad\mathfrak g=\bigoplus_j\mathfrak g_j\text{ with each }\mathfrak g_j\text{ simple}.}
$$
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