TITLE: The Killing forms of $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic (as real Lie algebras)
QUESTION [5 upvotes]: Show that $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic as  real Lie algebras.

I calculated that both of them have the Killing form $$K(x,y)=2n \,\textrm{tr}(xy) .$$ I got the hint that I should consider the signature of both Killing forms. Can you tell me how I can use this hint?

REPLY [3 votes]: One can also show that $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic without using the Killing form. One algebra has subalgebras of a certain dimension, the other one does not - see the following question:
Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic. 
Furthermore, one algebra is compact, the other one not (here one could use the Killing form, which is negative definite for one algebra, but not for the other).