Levi Decomposition of a real Lie algebra with codimension two semisimple Lie subalgebra.

Question

I am starting to learn some Lie theory and have a few questions about Levi decomposition in a specific case.

I have a real Lie algebra $\mathfrak{g}$ of dimension $n$ with a semi-simple Lie subalgebra $\mathfrak{k}$ of dimension $n-2$. Then let $\mathfrak{s}$ be the maximal semisimple subalgebra of $\mathfrak{g}$ containing $\mathfrak{k}$.

Then by the Levi decomposition we can write $$ \mathfrak{g} = \mathfrak{s} \oplus \mathfrak{r}$$ where $\mathfrak{r}$ is the radical of $\mathfrak{g}$.

So we have three possible cases:

1. The dimension of $\mathfrak{s} = n-2$, i.e. $\mathfrak{s} = \mathfrak{k}$,
2. The dimension of $\mathfrak{s} = n-1$
3. The dimension of $\mathfrak{s} = n$, i.e. $\mathfrak{s} = \mathfrak{g}$.

Are the second two cases possible?

Since semisimple Lie algebras have dimension at least $3$ it intuitively seems that one cannot have a codimension 1 (or two) semisimple subalgebra of a semisimple Lie algebra - but I am unsure how to prove this either way.

Any help or suggested references is appreciated, thanks.