We classify up to conjugation by [Formula: see text] (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of [Formula: see text] by means of an essentially geometric argument. This classification has already been established in [G. S. Alberti, L. Balletti, F. De Mari and E. De Vito, Reproducing subgroups of [Formula: see text]. Part I: Algebraic classification, J. Fourier Anal. Appl. 9(4) (2013) 651–682] without geometry, under a stricter notion of equivalence, namely, conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.