Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. If necessary, I would not mind making some assumptions regarding $\mathfrak g$, like assuming it is simple or semisimple.
Definition: A subalgebra $\mathfrak h \subset \mathbb C \mathfrak g$ is called elliptic if $\mathfrak h + \overline {\mathfrak h} = \mathbb{C}\mathfrak g$.
The name elliptic is used because there is a natural differential operator associated with the subalgebra $\mathfrak h$ that is elliptic if and only if $\mathfrak h$ is elliptic.
My question is: does it exists a proper semisimple elliptic subalgebra $\mathfrak h \subset \mathbb C \mathfrak g$?
EDIT: I know that, for example, if the dimension of $\mathfrak g$ is 3, then any proper elliptic Lie algebra will have dimension 2 and such subalgebra will never be semisimple. So, not all Lie algebras $\mathbb C \mathfrak g$ will have semisimple proper elliptic Lie algebras. I would not mind assuming some extra hypothesis on $\mathfrak g$. Actually, if I can find any example of semisimple proper elliptic Lie algebra I will be very happy.
Thank you very much for any help.
No, such subalgebra need not exist. For instance, take ${\mathfrak g}= o(3)$, ${\mathbb C}{\mathfrak g}\cong sl(2, {\mathbb C})$. However, $sl(2, {\mathbb C})$ contains no proper semisimple subalgebras (simply because 3 is the smallest dimension of a complex simple algebra).