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Wikipedia's entry for Euler's equation states:

In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. Their general form is:

$$I\dot{\omega}+\omega\times (I \omega) = M.$$

Is the bolded part really necessary? That would mean that the equation only works when $I$ is diagonal, and that doesn't seem right to me.

Also, just to be certain, both $\omega$ and $M$ in the equation are expressed in the fixed body frame, right?

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You're misreading. (Alternatively, the text in that wikipedia article is not exactly well written). Euler's equations, which is the subject of that wikipedia article, are

\begin{aligned}I_{1}{\dot {\omega }}_{{1}}+(I_{3}-I_{2})\omega _{2}\omega _{3}&=M_{{1}}\\I_{2}{\dot {\omega }}_{{2}}+(I_{1}-I_{3})\omega _{3}\omega _{1}&=M_{{2}}\\I_{3}{\dot {\omega }}_{{3}}+(I_{2}-I_{1})\omega _{1}\omega _{2}&=M_{{3}}\end{aligned}

This set of equations is valid only in the case that the inertia tensor is diagonal (in other words, when the body axes are parallel to the body's principal axes of inertia). The general form, valid for any body fixed frame, is

$$\mathrm{I}\dot{\boldsymbol{\omega}} + \boldsymbol{\omega}\times\left(\mathrm{I} \boldsymbol \omega\right) = \boldsymbol M$$

Also, just to be certain, both $\boldsymbol \omega$ and $\boldsymbol M$ in the equation are expressed in the fixed body frame, right?

Correct. Being pedantically correct, $\boldsymbol \omega$ is the angular velocity of the object in question with respect to an inertial frame but expressed in body frame coordinates, and $\boldsymbol M$ is the external non-fictitious torque on the body, once again expressed in body frame coordinates. One way of looking at the $\boldsymbol{\omega}\times\left(\mathrm{I} \boldsymbol \omega\right)$ term is that it is a fictitious torque, the torque analogy of the fictitious centrifugal, Coriolis, and Euler forces.

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