Whitehead's lemma

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Whitehead's lemma states that a matrix of the form \begin{bmatrix} u & 0 \\ 0 & u^{-1} \end{bmatrix} is equivalent to identity by elementary transformations:

\begin{bmatrix} u & 0 \\ 0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}).

Here, eij(s) indicates a matrix whose diagonal block is 1 and ijth entry is s.

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