= Symplectic group over a field
{title2=$Sp(V,B)$}
For a finite-dimensional <vector space> over any <field> with a <nondegenerate> <alternating bilinear form> $B$, the <symplectic group over a field> consists of <invertible linear maps> preserving $B$. In a <symplectic basis> its matrices satisfy $g^{\mathsf T}Jg=J$. Matching two <symplectic bases> gives a unique such <linear map>, and a <symplectic basis> exists by successively splitting off paired vectors. This includes arbitrary-characteristic and infinite-field versions, beyond the real <Lie group> and the <symplectic group over a finite field>.
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