Simple Lie algebra over ${\bf R}$

Question

As far as I know classification of simple Lie algebra over ${\bf C}$ in ${\rm gl}\ (n, {\bf C})$ is done.

And note that $$ {\bf R}^3_\wedge \otimes {\bf C} = {\rm sl}\ (2,{\bf C})$$

(all of $3$-dimensional simple Lie algebra over ${\bf R}$ are ${\bf R}^3_\wedge ,\ {\rm sl}\ (2,{\bf R})$)

So

question : Classification of simple Lie algebra over ${\bf R}$ in ${\rm gl}\ (n, {\bf R})$ is done ?

Answer 1

Not only is it done, but there is a quick proof (A. Knapp, 1996).