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.
Does this do you? Magma V2.1111 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; > m2 := Adjoint(m1)*Adjoint(y)*m1*y; > I := ideal<REltseq(m2d),Determinant(y)1>; > Groebner(I); > I; Ideal of Polynomial ring of rank 9 over Rational Field Graded Reverse Lexicographical Order Variables: x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8], x[9] Groebner basis: [ 1 ] 

