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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 $x=I$. |
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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 $x=I$. |
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