In February, with the images of the violent insurrection in Washington still fresh in the minds of Americans, newly confirmed Defense Secretary Lloyd Austin took the unprecedented step of signing a memo directing commanding officers across the military to institute a one-day stand-down to address extremism within the nation's armed forces.

The stand-down came in response to the participation and the subsequent arrests of several veterans and at least one active duty service member, who along with thousands of supporters of former President Donald Trump on Jan. 6, stormed the U.S. Capitol in a...