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We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (moreprecisely, block diagonal symplectic matrices) all the semidirect productsinside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means ofan essentially geometric argument. This classification has already beenestablished without geometry, under a stricter notion of equivalence, namelyconjugation by arbitrary symplectic matrices. The present approach might beuseful in higher dimensions and provides some insight.

Geometric classification of semidirect products in the maximal parabolic subgroup of $\operatorname{Sp}(2,\mathbb{R})$