About solvable algebras and solvable structures

See also about Lie algebras and involutive sets of vector fields.

1. Lie algebras are defined like vector spaces with a binary operation $[-,-]$, bla, bla, bla. The result of the bracket has to be expressed as linear combinations with coefficients in $\mathbb{R}$, of course. Some Lie algebras are finite dimensional and other infinite dimensional.
2. Some Lie algebras are solvable, in the sense that the derived series arrives at 0 in a finite number of steps. As far as I know, they can be finite or infinite dimensional.
3. An example of an infinite dimensional Lie algebra: vector fields on a manifold $\mathfrak{X}(M)$. If we fix a distribution $\mathcal{Z}\subseteq \mathfrak{X}(M)$ in $M$, there is a (possibly infinite dimensional) Lie subalgebra consisting of all the symmetries of the distribution $\text{Sym}(\mathcal{Z})$. Remember: they are vector fields $X$ such that