Here they say that the columns of a symplectic matrix form a symplectic basis, however, the link for the basis entry is empty. What is the definition of a symplectic basis?
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A basis $(u_1,\ldots,u_n,v_1,\ldots,v_n)$ of a symplectic vector space $(V,\omega)$ is symplectic if $\omega(u_i,v_i)=1$, $\omega(v_i,u_i)=-1$, and all other pairings between basis vectors are zero. Such a basis is often also called a Darboux basis.