Simultaneous conjugation in quaternion algebras (Q1320080)

The authors prove the following general theorem: Let \(Q\) be a quaternion algebra over a field \(K\) of characteristic \(\neq 2\). Let \(I = A_ 0,A_ 1,\dots,A_ n\), \(I = B_ 0,B_ 1,\dots,B_ n \in Q\) be such that \(\text{red.tr}(A_ iA_ j) = \text{red.tr}(B_ iB_ j)\) for \(0 \leq i,j \leq n\). The following are equivalent: (i) \(A_ i \to B_ i\) defines a \(K\)-vector space isomorphism between the spaces they generate; (ii) there exists a unit \(C \in Q\) with \(C A_ i C^{-1} = B_ i\) all \(i\) or \(= \overline{B}_ i\) for all \(i\). This follows easily from Witt's theorem on extending isometries. It can be applied in particular to the case where \(Q = K^{2\times 2}\) when the additional condition \(\text{red.tr }[A_ 1,A_ 2] \neq 2\) implies the subspaces in (i) are non-degenerate so that (i) automatically holds. Furthermore, if \(K\) is algebraically closed, then \(C\) can be taken to lie in \(SL(2,K)\).