Random generation of direct sums of finite non-degenerate subspaces

Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subspace of V, such that U∩U′=0 and U⊕U′ is non-degenerate (the sum U⊕U′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim⁡(U) and dim⁡(U′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U′) of non-degenerate subspaces and certain pairs (g,g′)∈G2 of group elements where U=im(g−1) and U′=im(g′−1).