The last execution in the Army was April 13, 1961, U.S. Army Private John A. Bennett was hanged after being convicted of rape and attempted murder of an 11 year old girl.
In this case, the crime was committed by a soldier, against soldiers, and on a military installation. Clearly the Army has jurisdiction, and murder is a violation of Article 118 of the UCMJ.
At present the are 9 service members on death row in the US Disciplinary Barracks at Fort Leavenworth. No service member can be executed unless the President personally confirms the death penalty.
No danger of that happening with Obambi. He'll probably give Hasan some kind of commendation.