= Solution
The <derived series of a Lie algebra> is defined recursively by $\mathfrak g^{(0)}=\mathfrak g$ and $\mathfrak g^{(r+1)}=[\mathfrak g^{(r)},\mathfrak g^{(r)}]$, where the bracket denotes the <linear span> of all brackets of pairs of elements. A <solvable Lie algebra> is one whose <derived series of a Lie algebra> terminates at zero:
$$
\boxed{\mathfrak g\text{ is solvable}\iff\mathfrak g^{(m)}=0\text{ for some }m\geq0.}
$$
The <solvable radical> $\operatorname{rad}\mathfrak g$ of a finite-dimensional <Lie algebra> is its largest solvable <ideal of a Lie algebra>. A <semisimple Lie algebra> has no nonzero solvable <ideals of a Lie algebra>, equivalently
$$
\boxed{\mathfrak g\text{ is semisimple}\iff\operatorname{rad}\mathfrak g=0.}
$$
These are definitions; identifying semisimplicity with a direct sum of simple ideals additionally uses the structure theorem in <characteristic zero>. The calculations and representation theorems below are over $\mathbb C$, as in the complex representation setting of this paper.
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