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In this note, we describe the geometry of the quaternionic Heisenberg groupsfrom a Riemannian viewpoint. We show, in all dimensions, that they carry analmost $3$-contact metric structure which allows us to define the metricconnection that equips these groups with the structure of a naturally reductivehomogeneous space. It turns out that this connection, which we shall call thecanonical connection because of its analogy to the $3$-Sasaki case, preservesthe horizontal and vertical distributions and even the quaternionic contactstructure of the quaternionic Heisenberg groups. We focus on the$7$-dimensional case and prove that the canonical connection can also beobtained by means of a cocalibrated $G_2$ structure. We then study thespinorial properties of this group and present the noteworthy fact that it isthe only known example of a manifold which carries generalized Killing spinorswith three different eigenvalues.

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces