A note on Kottwitz's invariant \(e(G)\) (Q1271127)

Let \(G\) be a connected reductive group over a field \(k\) with \(\text{char} (k)\neq 2\). \textit{R. E. Kottwitz} [Trans. Am. Math. Soc. 278, 289-297 (1983; Zbl 0538.22010)] has defined an invariant \(e(G)\in H^2(k, \mu_2)\), \(\mu_2=\langle\pm 1\rangle\). Let \(W(k)\) be the Witt ring of nondegenerate quadratic forms over \(k\), and let \(I\) be the ideal of even rank forms. Then \(I^2/I^3 \cong H^2(k,\mu_2)\). Hence \(e(G)\) can be regarded as element in \(I^2/I^3\). This paper gives an interpretation of \(e(G)\) in terms of quadratic forms. An invariant \(e'(G)\) is defined in terms of the Killing form of \(G\) and its quasi-split inner form and it is shown that \(e(G)=e'(G)\).