\chapter{Solvable Lie groups and Lie algebrae}

Related to a condition on the lie bracket for the lie algebra




\section{Solvable Lie algebrae}
A \emph{solvable lie algebra} is a Lie algebra \mathfrak{g} such that its chain of derived algebras eventually reach the trivial Lie algebra.


For finite dimension Lie algebrae, there exists a solvable ideal $I$ such that all solvable ideals are subsets of $I$

This result allows for a strict notion of a "maximal solvable ideal" in such spaces; we call it the \emph{radical} of the space.
Radical
$\mathrm{rad}(\mathfrak{g})$
\subsection{Nilpotent Lie algebrae}
\subsection{Simple Lie algebrae}



\section{Classification of simple Lie algebrae}

$\mathfrak{sl}(n,\mathbb{C}) , \mathfrak{so}(n,\mathbb{C}) , \mathfrak{sp}(2n,\mathbb{C})$
$\mathfrak{e}_6 , \mathfrak{e}_7 , \mathfrak{e}_8 , \mathfrak{f}_4 , \mathfrak{g}_2$








\section{Root systems}