Euler's equations for a rigid body express the change in angular momentum with respect to the rotating system, but the torques are expressed for the inertial system. For example, see the explanations in Goldstein, Classical Mechanics or in Symon, Mechanics. This was confusing to me in my first Mechanics physics course.

Euler's equations for a rigid body express the change in angular momentum in the inertial frame $\vec L$ with respect to the non-inertial rotating frame coordinates. See relationships (2) and (3) and the associated discussion in the answer by @David Hammen. You use relationship (3) in David's relationship (1) which is in the inertial system, where no fictitious torques are present.

For example, see the explanations in Goldstein, Classical Mechanics or in Symon, Mechanics. This was confusing to me in my first Mechanics physics course.