I have the following doubt. Say that $\mathfrak g$ is a semisimple Lie algebra, $\mathfrak k$ a reductive subalgebra, and suppose further that any Cartan subalgebra in $\mathfrak k$ is a Cartan subalgebra in $\mathfrak g$. The reasoning that leads me to say that $\frak k$ is semisimple as well is the following: since $\frak k$ is reductive, we can write it as a direct sum \begin{equation} \frak k = \frak s + \frak l \end{equation} with $\frak s$ Abelian and $\frak l$ semisimple. As far as I understand, $[\frak s,\frak l]=0$.
Now, $\frak s$ can be made to sit in some Cartan subalgebra of $\frak k$, right? And I would say that we can express $\frak l$ via a system of subroots of the root system of $\frak g$ - the bottom line is that I don't see how $[\frak s,\frak l]=0$ could hold. But also, this reasoning is far from crystal clear.
So I'd be happy to receive feedback, counterexamples, references.