Let $\mathfrak{g}$ be a Lie algebra (finite or infinite dimensional). By using the commutator subalgebra we can construct the _derived series_ of $\mathfrak{g}$
$$ \mathfrak{g} \geq[\mathfrak{g}, \mathfrak{g}] \geq[[\mathfrak{g}, \mathfrak{g}],[\mathfrak{g}, \mathfrak{g}]] \geq[[[\mathfrak{g}, \mathfrak{g}],[\mathfrak{g}, \mathfrak{g}]],[[\mathfrak{g}, \mathfrak{g}],[\mathfrak{g}, \mathfrak{g}]]] \geq \ldots $$If this series arrives to 0 in a finite number of steps it is said that $\mathfrak{g}$ is solvable.
Equivalently:
Definition (@olver86). A Lie algebra $\mathfrak{g}$ is a solvable Lie algebra if there exists a chain of subalgebras
$$ \{0\} = \mathfrak{g}_0 \subseteq \mathfrak{g}_1 \subseteq \mathfrak{g}_2 \subseteq \ldots \subseteq \mathfrak{g}_{r-1} \subseteq \mathfrak{g}_r = \mathfrak{g} $$such that for each $k$, $\dim \mathfrak{g}_{k-1} = k$ and $\mathfrak{g}_{k-1}$ is a normal subalgebra of $\mathfrak{g}_k$:
$$ [\mathfrak{g}_{k-1},\mathfrak{g}_k]\subseteq \mathfrak{g}_{k-1}. $$The requirement for solvability is equivalent to the existence of a basis $\{v_1, \ldots, v^r\}$ of $\mathfrak{g}$ such that
$$ [v_i, v_j] = \sum_{k}^{j-1} c_{ij}^k v_k $$whenever $i < j$.
$\blacksquare$
Other characterizations:
Theorem
A Lie algebra $\mathfrak{g}$ is solvable if and only if there exists a sequence of subalgebras
$$ \{0\}=a^{0} \subset a^{1} \subset \ldots \subset a^{k}=\mathfrak{g} $$such that $a^{i}$ is an ideal of $a^{i+1}$ and the quotient $a^{i+1}/a^i$ is abelian.$\blacksquare$
See Ruiz_2014, page 7.
Theorem
Let $\mathfrak{g}$ be a solvable Lie algebra. There exists a sequence of ideals of $\mathfrak{g}$
$$ \{0\} \subset L^{1} \subset \ldots \subset L^{n}=\mathfrak{g} $$such that $\mbox{dim}(L^i)=i$ for $i=1,\ldots,n$.$\blacksquare$
See Ruiz_2014, page 8.
It can be shown that a connected Lie group is solvable if and only if its Lie algebra is solvable.
On the other hand, if $\mathfrak{g}$ is a finite dimensional Lie algebra, there exists a unique maximal solvable ideal called the radical of a Lie algebra.
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Author of the notes: Antonio J. Pan-Collantes
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