Lie's theorem
If you have several commuting matrices you can diagonalize them all at once in a common basis. Solvable Lie algebras are not commutative, but they are close enough to be tamed simultaneously: there is a single basis in which every element of the algebra becomes upper-triangular. Lie's theorem makes this precise and is the solvable-case analogue of Engel's nilpotent-case theorem.
Let L be a solvable Lie algebra of operators on a finite-dimensional nonzero vector space V over an algebraically closed field of characteristic zero. Then the operators in L have a common eigenvector in V. Iterating, V admits a complete flag 0 = V_0 < V_1 < ... < V_n = V stabilized by all of L, so in a suitable basis every element of L is upper-triangular.
Both hypotheses matter. Without algebraic closure there may be no eigenvector at all (a rotation has no real eigenvector). In positive characteristic the theorem fails: there are solvable Lie algebras of operators over fields of characteristic p with no common eigenvector. A corollary in characteristic zero is that [L, L] is then nilpotent, since its operators are strictly upper-triangular.
Over C, the solvable Lie algebra of all upper-triangular 2-by-2 matrices already acts on C^2 with the common eigenvector e_1 = (1, 0): every upper-triangular matrix [a, b; 0, d] sends e_1 to a.e_1. Lie's theorem guarantees this in general.
Upper-triangular matrices share the eigenvector e_1, the simplest case of Lie's theorem.
Lie's theorem describes the representation (it triangularizes operators); Engel's describes the abstract algebra (it concludes nilpotence from ad-nilpotence). They are dual companions, easily confused.