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We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace.
Watt, C., Brady, T. : A Partial Order on the Orthogonal Group. Communications in Algebra, Vol. 30, (2002), no. 8, pp 3749–3754