solvable Lie algebra

Let $\\mathfrak{g}$ be a Lie algebra. The lower central series of $\\mathfrak{g}$ is the filtration of subalgebras

\\mathcal{D}_{1}\\mathfrak{g}\\supset\\mathcal{D}_{2}\\mathfrak{g}\\supset\\mathcal{D%}_{3}\\mathfrak{g}\\supset\\cdots\\supset\\mathcal{D}_{k}\\mathfrak{g}\\supset\\cdots

of $\\mathfrak{g}$ , inductively defined for every natural number $k$ as follows:

	$\\displaystyle\\mathcal{D}_{1}\\mathfrak{g}$	$\\displaystyle:=$	$\\displaystyle[\\mathfrak{g},\\mathfrak{g}]$
	$\\displaystyle\\mathcal{D}_{k}\\mathfrak{g}$	$\\displaystyle:=$	$\\displaystyle[\\mathfrak{g},\\mathcal{D}_{k-1}\\mathfrak{g}]$

The upper central series of $\\mathfrak{g}$ is the filtration

\\mathcal{D}^{1}\\mathfrak{g}\\supset\\mathcal{D}^{2}\\mathfrak{g}\\supset\\mathcal{D%}^{3}\\mathfrak{g}\\supset\\cdots\\supset\\mathcal{D}^{k}\\mathfrak{g}\\supset\\cdots

defined inductively by

	$\\displaystyle\\mathcal{D}^{1}\\mathfrak{g}$	$\\displaystyle:=$	$\\displaystyle[\\mathfrak{g},\\mathfrak{g}]$
	$\\displaystyle\\mathcal{D}^{k}\\mathfrak{g}$	$\\displaystyle:=$	$\\displaystyle[\\mathcal{D}^{k-1}\\mathfrak{g},\\mathcal{D}^{k-1}\\mathfrak{g}]$

In fact both $\\mathcal{D}^{k}\\mathfrak{g}$ and $\\mathcal{D}_{k}\\mathfrak{g}$ are ideals of $\\mathfrak{g}$ , and $\\mathcal{D}^{k}\\mathfrak{g}\\subset\\mathcal{D}_{k}\\mathfrak{g}$ for all $k$ . The Lie algebra $\\mathfrak{g}$ is defined to be nilpotent if $\\mathcal{D}_{k}\\mathfrak{g}=0$ for some $k\\in\\mathbb{N}$ , and solvable if $\\mathcal{D}^{k}\\mathfrak{g}=0$ for some $k\\in\\mathbb{N}$ .

A subalgebra $\\mathfrak{h}$ of $\\mathfrak{g}$ is said to be nilpotent or solvable if $\\mathfrak{h}$ is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of $\\mathfrak{g}$ , since every ideal of $\\mathfrak{g}$ is also a subalgebra.