  We establish a few properties of eigenvalues and eigenvectors of the
quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex
Ginibre case. We first recover a version of Kostlan's theorem that was already
noticed by Rider: the set of the squared radii of the eigenvalues is
distributed as a set of independent gamma variables. Our proof technique uses
the De Bruijn identity and properties of Pfaffians; it also allows to prove
that the high powers of these eigenvalues are independent. These results extend
to any potential beyond the Gaussian case, as long as radial symmetry holds;
this includes for instance truncations of quaternionic unitary matrices,
products of quaternionic Ginibre matrices, and the quaternionic spherical
ensemble.
  We then study the eigenvectors of quaternionic Ginibre matrices. The angle
between eigenvectors and the matrix of overlaps both exhibit some specific
features that can be compared to the complex case. In particular, we compute
the distribution and the limit of the diagonal overlap associated to an
eigenvalue that is conditioned to be zero.
