Linear relations in Eichler orthogonal transformations (Q1335098)

Let \(A\) be a commutative ring in which 2 is invertible. In [J. Algebra 86, 159-180 (1984; Zbl 0537.10013)], \textit{R. A. Rao} proved a splitting lemma for Eichler orthogonal transformations, tacitly assuming that these transformations satisfy a simple linear relation. The aim of this note is to show that they satisfy a slightly more complicated linear relation and to prove a version of the splitting lemma, crucial for the proof of the main theorem of [loc. cit.]. The result also yields a simpler proof of a splitting theorem of \textit{I. Bertuccioni} [\(K\)-Theory 1, 185-196 (1987; Zbl 0631.13011)].