---- The key step in proving [[Engel's Theorem]] was to show that if a [[general linear Lie algebra|general linear]] [[Lie subalgebra]] $\mathfrak{g} \subset \mathfrak{gl}(V)$ consists of [[nilpotent linear operator|nilpotent linear operators]], then there is a [[basis]] of $V$ with respect to which $\mathfrak{g}$ consists of strictly [[upper-triangular matrix|upper triangular matrices]], $\mathfrak{g} \subset \mathfrak{n}_{n}$. Lie's Theorem says that something similar holds for solvable subalgebras and the [[upper-triangular matrix|upper triangular]] matrices $\mathfrak{b}_{n}$. ^94e8cf > [!theorem] Theorem. ([[Lie's Theorem]]) > Let $\mathbb{F}$ be a [[field]] that is [[algebraically closed]] and has [[characteristic of a field|characteristic zero]] (e.g., $\mathbb{C}$). Let $V$ be an $\mathbb{F}$-[[vector space]], and let $\mathfrak{g} \subset \mathfrak{gl}(V)$. > If $\mathfrak{g}$ is [[derived and central series of a Lie algebra|solvable]], then its elements are [[simultaneously diagonalizable|simultaneously]] [[upper-triangular matrix|upper triangularizable]], i.e., there exists a [[basis]] of $V$ such that $\mathfrak{g} \subset \mathfrak{b}_{n}$. > Note that the converse of course holds, since $\mathfrak{b}_{n}$ is [[derived and central series of a Lie algebra|solvable]]. ^theorem > [!proposition] Corollary. > If $\mathfrak{g} \subset \mathfrak{gl}(V)$ is [[derived and central series of a Lie algebra|solvable]], then its elements have a simultaneous [[eigenvector]]. ^proposition > [!proof]- Proof. ([[Lie's Theorem]]) > Similar to that of [[Engel's Theorem]]; omitted in our course. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```