Answer
The mosaic dataset has several layer properties: on the main mosaic dataset group layer, on the boundary sub-layer, on the footprint sub-layer, and on the image sub-layer (possibly others when available).
The layer properties that are opened through the Image Analysis window are the same opened from the image sub-layer in the Table of Contents.
On acceleration, the mosaic dataset layer in the Table of Contents is also the same as the image sub-layer on the un-accelerated mosaic dataset.