I was trying to prove that, $[SL_n(\Bbb R), SL_n(\Bbb R)]=SL_n(\Bbb R)$
My attempt:
$[SL_n(\Bbb R), SL_n(\Bbb R)]\subset SL_n(\Bbb R)$ is trivial, trying to prove the other inclusion.
So it's enough to show that given any $A \in SL_n(\Bbb R)$ $\exists B,P \in SL_n(\Bbb R)$ such that, $A=BPB^{-1}P^{-1}$ .
In fact, I proved that $SL_n(\Bbb R)$ is generated by elementary matrices of the form $\{I_n+\lambda e_{ij} | \lambda\in \Bbb R , i \ne j\}$.
So the problem can be reduced to :
Given an elementary matrix of the above form say, $E=I_n+\lambda e_{ij}$ find , $B,P \in SL_n(\Bbb R)$ such that $E=BPB^{-1}P^{-1}$
Thanks in advance for help!