# Invertible matrices satisfying $[x,y,y]=x$ (take 2).

This is a simpler version of this question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\ 0& 1 & 0\\\ 0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with complex entries and $\det(y)=1$ such that $[x,y,y]=x$? Here $[a,b]=a^{-1}b^{-1}ab$, $[a,b,c]=[[a,b],c]]$? Perhaps somebody with a good Groebner basis software can check it.

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Does this do you?

Magma V2.11-11    Sun Jul 24 2011 20:03:50 on sevilla  [Seed = 2330466759]
Type ? for help.  Type -D to quit.
> R<[x]> := PolynomialRing(Rationals(),9,"grevlex");
> y := Matrix(3,x);
> d := DiagonalMatrix(R,[2,1,1/2]);
> m1 := d^-1*Adjoint(y)*d*y;