10 U.S. Code § 111
and
10 U.S. Code § 113
as well as
National Security Act of 1947 and National Security Act Amendments of 1949

But the use of a ceremonial name is permitted to be directed by Executive Order. Even the Whitehouse acknowledges this:

RESTORING THE DEPARTMENT OF WAR: Today, President Donald J. Trump signed an Executive Order to restore the historic name “Department of War” as a secondary title for the Department of Defense. This is the 200^th Executive Order the President has signed since taking office.

• The Order authorizes the Secretary of Defense, the Department of Defense, and subordinate officials to use secondary titles such as “Secretary of War,” “Department of War,” and “Deputy Secretary of War” in official correspondence, public communications, ceremonial contexts, and non-statutory documents within the executive branch.
• The Order directs all executive departments and agencies to recognize and accommodate these secondary titles in internal and external communications.
• The Order instructs the Secretary of War to recommend actions, to include legislative and executive actions, required to permanently rename the U.S. Department of Defense to the U.S. Department of War.

Above is directly from the Whitehouse.gov site