On Computer A, I have a public and private key for my github account. When I use this machine directly, using a connected keyboard and display, I can do all the git remote repo commands, like git pull and git push, as well as ssh -vv firstname.lastname@example.org. All this works and uses the proper ssh keypair.
But if I log into Computer A from Computer B, and then try the same stuff as above, I can't seem to get the same ssh keys to work with GitHub's servers. There is absolutely no difference here, except that I am controlling Computer A via an ssh connection, rather than directly via the computer's keyboard.
What's going on here exactly, and how can I learn more about it?