I could not find any satisfactory answer online so I ask it here. The problem is the following:
I have a linear complex Lie algebra $\mathfrak{g}$ which is algebraic $\textit{i.e.}$ the Lie algebra of a complex linear algebraic group $\mathbb{G}(\mathbb{C})$. Are necessarily all real forms of $\mathfrak{g}$ also algebraic (the Lie algebra of a real algebraic group)? In my case $\mathfrak{g}$ is algebraic because it is semisimple but I'm not sure that this information is relevant for the question.
Thank you!