Ignore this, it is wrong
I migth miss something simple, but
$[a,b]^n=[a,b]$ for all $n$ hence $x^2=[x,y,y]^2=[x,y,y]=x$.
Since $x$ is invertible and $x^2=x$ it follows $x=I$.
using this it is easy to show that $[x,y,y]=x$ for x,y invertible if and only if $x=I$.