Let $\mathfrak{g}$ be the Lie algebra of a compact semisimple Lie group and $\mathfrak{g}_{\mathbb{C}}$ its complexification. If $\mathfrak{h}_{\mathbb{C}}$ is a complex semisimple subalgebra of $\mathfrak{g}_{\mathbb{C}}$, is $\mathfrak{h}:=\mathfrak{h}_{\mathbb{C}}\cap\mathfrak{g}$ a compact real form for $\mathfrak{h}_{\mathbb{C}}$?
In particular, I am interested in $\mathfrak{sl}(2,\mathbb{C})$-subalgebras of $\mathfrak{g}_{\mathbb{C}}$ and my question is if they all restrict to $\mathfrak{su}(2)$-subalgebras of $\mathfrak{g}$.
Clarification of the terminology: An "$\mathfrak{sl}(2,\mathbb{C})$-subalgebra of $\mathfrak{g}_{\mathbb{C}}$" is a subalgebra of $\mathfrak{g}_{\mathbb{C}}$ which is isomorphic $\mathfrak{sl}(2,\mathbb{C})$. Similarly for $\mathfrak{su}(2)$-subalgebras of $\mathfrak{g}$.