Editing Lie algebra (section)

==== Example ====
The subspace <math>\mathfrak{t}_n</math> of diagonal matrices in <math>\mathfrak{gl}(n,F)</math> is an abelian Lie subalgebra. (It is a [[Cartan subalgebra]] of <math>\mathfrak{gl}(n)</math>, analogous to a [[maximal torus]] in the theory of [[compact Lie group]]s.) Here <math>\mathfrak{t}_n</math> is not an ideal in <math>\mathfrak{gl}(n)</math> for <math>n\geq 2</math>. For example, when <math>n=2</math>, this follows from the calculation:
<blockquote><math>\begin{align}
\left[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix},
\begin{bmatrix}
x & 0 \\
0 & y
\end{bmatrix}
\right] &= \begin{bmatrix}
ax & by\\
cx & dy \\
\end{bmatrix} - \begin{bmatrix}
ax & bx\\
cy & dy \\
\end{bmatrix} \\
&= \begin{bmatrix}
0 & b(y-x) \\
c(x-y) & 0
\end{bmatrix}
\end{align}</math></blockquote>
(which is not always in <math>\mathfrak{t}_2</math>).

Every one-dimensional linear subspace of a Lie algebra <math>\mathfrak{g}</math> is an abelian Lie subalgebra, but it need not be an ideal.