In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given:
$$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$
is symplectic if and only if:
$$A^TD-C^TB=I, A^TC=C^TA, D^TB=B^TD$$
but it seems wrong, since, for example:
$$\left( \begin{array}{ccc} 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right)$$
is symplectic but doesn't satisfy the conditions. Or have I mixed everything up?
EDIT: this is crazy talk. That matrix isn't symplectic! (not for the form defined in Wikipedia or MathPlanet.