from what I know this is a portrait of Poor Bear, Kiowa-Apache chief, and Caddo Jake, both part of the Southern Plains delegation in 1863 to Washington. Which leads me to the conclusion that the portrait of Satank, which is published in numerous books and all across the internet is perhaps not him but also Poor Bear. At least this is stuff for discussion (I remember I chatted with Grahame about it in another thread). William S. Soule selled the portrait as one of his own photographs, but perhaps he just copied it from one of the earlier delegation pictures.
I know that would be kind of disappointing, because it would mean there is no authentic portrait of Satank.
There is one clue here. The Moccasins, they are Kiowa Mocs.
and thank you for your comment. Do you mean that it would disqualify Poor Bear? I think the Plains-/Kiowa-Apaches had become very close to the Kiowas in the 19th century. But I must admit I´m not an expert on Indian material culture.