Does it exist a Lie algebra $\mathfrak{g}$ that is reductive but if we consider the inclusion of Lie agebras $\mathfrak{g} \subset \mathfrak{h}$ then $\mathfrak{g}$ is not reductive in $\mathfrak{h}?$



Reductive Lie algebra
A reductive Lie algebra $L$ is the direct sum of a semisimple
Lie algebra $L\_1$ and an abelian Lie algebra $L\_2$. Let's consider
the case where $L\_2$ is one-dimensional.
We can embed $L$ into a larger Lie algebra $L=L\_1\oplus L\_2'$
by embedding $L\_2$ into $L\_2'$. Let $L\_2'$ be the two-dimensional
Lie subalgebra
$$\left(\begin{array}{cc}
\*& \*\\\
0& 0
\end{array}
\right)$$
of $\mathfrak{gl}(\mathbf{C})$
and
$$L\_2=\left(\begin{array}{cc}
0& \*\\\
0& 0
\end{array}
\right).$$
Then $L\_2$ does not act semisimply on $L\_2'$, so $L$ does not
act semisimply on $L'$.


