The Schur index of a central simple algebra \(A\) over a field \(F\) is the degree of the division algebra which is Brauer-equivalent to \(A\). The purpose of this paper is to prove formulas describing how the Schur index of a central simple algebra is reduced when scalars are extended to the function field of certain varieties of ideals in central simple algebras (i.e. (generalized) Brauer-Severi varieties) and (Weil) transfers of such varieties. Index reduction formulas were first proven by \textit{A. Schofield} and \textit{M. Van den Bergh} [Trans. Am. Math. Soc. 333, No. 2, 729-739 (1992; Zbl 0778.12004)] for function fields of Brauer-Severi varieties and by \textit{A. Blanchet} [Commun. Algebra 19, No. 1, 97-118 (1991; Zbl 0717.16014)]. In the present paper, a clever presentation of the relevant function fields (up to stable isomorphism) is given in terms of twisted group algebras, which allows the author to derive the various index reduction formulas from Moody's induction theorem for \(G_ 0\) of certain infinite groups.