SecDef is not supposed to be SuperChairman of the Joint Chiefs of Staff. Because the Honorable Secretary (retired General) was already one of the highest ranking military members, it would be difficult to say when the military ends and civilian begins. By the way, I believe it is a 7 year retirement rule. Because General Mattis is a retired four star, the same rank as CJCS General Dunford, it might make for conflict if there are disagreements.
A little background from Wiki:
The Chairman of the Joint Chiefs of Staff is, by law, the highest-ranking military officer of the United States Armed Forces, and the principal military adviser to the President of the United States. He leads the meetings and coordinates the efforts of the Joint Chiefs of Staff, comprising the chairman, the Vice Chairman of the Joint Chiefs of Staff, the Chief of Staff of the United States Army, the Commandant of the United States Marine Corps, the Chief of Naval Operations, the Chief of Staff of the United States Air Force, and the Chief of the National Guard Bureau. The Joint Chiefs of Staff have offices in The Pentagon. The chairman outranks all respective heads of each service branch, but does not have command authority over them, their service branches or the Unified Combatant Commands. All combatant commanders receive operational orders directly from the Secretary of Defense.
The current chairman is General Joseph Dunford, USMC, who began his term on October 1, 2015.
On 20 July 1942, Navy Fleet Admiral William D. Leahy became the Chief of Staff to the Commander in Chief of the Army and Navy (20 July 1942 – 21 March 1949). He was not technically the Chairman of the Joint Chiefs of Staff. Leahy’s office was the precursor to the post of Chairman of the Joint Chiefs of Staff. That post was established and first held by General of the Army Omar Bradley in 1949.
The position of Vice Chairman of the Joint Chiefs of Staff was created by the Goldwater-Nichols Act of 1986. The vice-chairman is a four-star-general or admiral and, by law, is the second highest-ranking member of the U.S. Armed Forces (after the Chairman of the Joint Chiefs of Staff). In the absence of the chairman, the vice-chairman presides over the meetings of the Joint Chiefs of Staff. He may also perform such duties as the chairman may prescribe. It was not until the National Defense Authorization Act in 1992 that the position was made a full voting member of the JCS.
The current vice-chairman is General Paul J. Selva, USAF.
Senior Enlisted Advisor to the Chairman
The Senior Enlisted Advisor to the Chairman of the Joint Chiefs of Staff (SEAC) advises on all matters concerning joint and combined total force integration, utilization, development, and helps develop noncommissioned officers related joint professional education, enhance utilization of senior NCOs on joint battle staffs, and support the chairman’s responsibilities as directed.
Command Sergeant Major William Gainey, USA, was the first SEAC, serving from October 1, 2005. The current SEAC is Command Sergeant Major John W. Troxell, U.S.A., who was sworn in by Joint Chiefs Chairman General Joseph Dunford on December 11, 2015, replacing Sergeant Major Bryan B. Battaglia, USMC.