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In this letter, we provide additional results on the expansion of wireless channels into spherical vector wave multipole modes. We specialize our results to multiple-input-multiple-output (MIMO) channels with Kronecker-product structure of the correlation matrix: 1) the Kronecker channel, and 2) the Keyhole channel. We show that the Kronecker or Keyhole structure of the multimode channel matrix implies the corresponding structure in the MIMO channel matrix. Moreover, we show that the separability of the joint probability distribution functions of the angle of arrival (AoA) and the angle of departure (AoD), i.e., the independence between the AoAs and the AoDs for co- and cross-polarized components results in a Kronecker-product structure of the full-correlation matrix of the multimode matrix for both the Kronecker and the Keyhole channels.