Solvable Lie algebra application

Besides Lie's theory of integration by quadrature for flows of vector fields, there is a completely different theory of solving rational coefficient linear differential equations by repeatedly integrating, taking logarithms, taking exponentials, and forming rational functions thereof: differential Galois theory. A nice introduction is: Camillo de Lellis, Il teorema di Liouville ovvero perche non esiste la primitiva di $e^{x^2}$. The upshot is that there is an invariant of any rational coefficient linear differential equation, its differential Galois group, and if the equation has general solution obtained by such repeated operations, then its differential Galois group is solvable.