Geodesic
Products of Positive Reflections in the Orthogonal Group
Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 484-499

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For G a group, S a subset of G which generates G, the length problem in G with respect to S is to find, for g ∈ G, the least integer r such that g can be written as the product of r elements of S. For G an orthogonal group Of(F) (here F is a field, and the elements of Of(F) preserve the quadratic form f) and S the set of reflections in Of(F) the length problem has been studied by E. Cartan [2], J. Dieudonné [4, 5], E. Ellers [7], P. Scherk [8], and others. In all of these investigations, however, the problem posed by requiring that S be a single conjugacy class of reflections in Of(F) has been ignored. And it is generally the case that the reflections in Of(F) fall into several conjugacy classes. 
Malzan, J. Products of Positive Reflections in the Orthogonal Group. Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 484-499. doi: 10.4153/CJM-1982-032-9
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