We say $\frak{h}$ is a characteristic ideal of a lie algebra $\frak{g}$ if $[\frak{h},\frak{g}]\subset\frak{h}$, and $D(\frak{h})\subset\frak{h}$ for every derivation $D\in Der(\frak{g})$.
The theorem states that if $\frak{h}$ is a characteristic ideal of a lie algebra $\frak{g}$, then every ideal of $\frak{h}$ is also an ideal in $\frak{g}$. Essentially, if we let $\frak{a}$ be an ideal in $\frak{h}$, we want to show that $[\frak{a},\frak{g}]\subset \frak{a}$, but I don't know what derivation $D\in Der(\frak{g})$ should be used?
If I am right, then let $x \in \mathfrak{g}$, $a\in\mathfrak{a}$, $\operatorname{ad}_x \colon \mathfrak{h}\rightarrow\mathfrak{h}$ defined by $\operatorname{ad}_x(h)=[x,h]$ is a derivation (Jacobi). This implies that $\operatorname{ad}_x(a)=[x,a]\in \mathfrak{a}$ since $\mathfrak{a}$ is a characteristic ideal of $\mathfrak{h}$. This implies that $\mathfrak{a}$ is an ideal of $\mathfrak{g}$.