Geodesic
Products of Reflections in the Group SO*(2n)
Canadian journal of mathematics, Tome 36 (1984) no. 2, pp. 300-326

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Let SO*(2n) be the group of quaternionic n × n matrices A satisfying A*JA = J, where J is a fixed skew-hermitian invertible matrix. An element R ∊ SO*(2n) is called a reflection if R 2 = In and R — In has rank one. We assume that n ≧ 2, in which case S*(2n) is generated by reflections. The length of A ∊ SO*(2n) is the smallest integer k(≧0) such that A can be written as A = R 1 R 2 ... Rk where R 1, ..., Rk are reflections. In this paper, for each A ∊ SO*(2n), we compute its length l(A). Set r(A) = rank (A — In ). Already in Section 3 we are able to show that the difference δ = l(A) – r(A) can take only three values 0, 1, or 2. The remainder of the paper deals with the problem of separating these three possibilities. The main results are stated in Section 4 and proved in Section 6. The intermediate Section 5 consists of a sequence of lemmas which are needed for the proof. Clearly l(A) depends only on the conjugacy class of A and the main results in Section 4 are stated in terms of conjugacy classes. For the description of conjugacy classes in SO*(2n) we refer the reader to [1]. The present paper relies heavily on our previous paper [5] where the analogous problem was solved for the groups U(p, q). It is worth remarking that only the various lemmas from that paper were used but not the main theorem. 
Djoković, Dragomir Ž.; Malzan, Jerry. Products of Reflections in the Group SO*(2n). Canadian journal of mathematics, Tome 36 (1984) no. 2, pp. 300-326. doi: 10.4153/CJM-1984-019-8
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