This question is based on the nomenclature and definitions in Structure and Interpretation of Classical Mechanics.
In Section 2.2, we start with a rotation $\mathscr{M}(q(t))$ where $q$ is the path describing the motion of the body. The rotation is later represented by the rotation matrix $\mathbf{M}$.
To paraphrase:
Let $\vec{\xi}_\alpha(t)$ be the vector to some constituent particle with the body in the orientation specified by q(t) given by $q(t)$, and $\vec{\xi'}_\alpha$ be the vector to the same constituent with the body in the reference orientation, then:
$$ \vec{\xi}_\alpha(t) = \mathscr{M}(q(t)) \vec{\xi'}_\alpha\tag{2.11} $$
Through a series of steps, it shows how the angular velocity vector consists of the components that can be extracted from the skew-symmetric matrix $D\mathbf{M} \mathbf{M}^\intercal$. Or
$$ \mathbf{\omega} = \mathscr{A}^{-1}( D\mathbf{M} \mathbf{M}^\intercal ) $$
$\mathscr{A}^{-1}$ is defined as the function that can extract the components of the angular velocity vector from an anti-symmetric matrix. In Eq. 2.20, it defines the derivative of the position of any constituent particle in the rigid body as:
$$ \dot{\vec{\xi}}_\alpha = \vec{\omega} \times \vec{\xi}_\alpha\tag{2.20} $$
While it is not explicitly mentioned anywhere, I assume that the angular velocity components mentioned above are in terms of the inertial reference frame. So after this, it goes on to state the following:
The components $\omega'$ of the angular velocity vector on the body axes are $\omega'= \mathbf{M}^\intercal \omega$, so
$$ \mathbf{\omega}' =\mathbf{M}^\intercal \mathscr{A}^{-1}( D\mathbf{M} \mathbf{M}^\intercal )\tag{2.21} $$
So here is my question:
In Eq. 2.11, $\mathbf{M}_\alpha$ seems to transform $\vec{\xi}$ from its reference frame components $\vec{\xi}'$ to its body-fixed frame components at time $t$. But in the last equation, angular velocity is transformed from reference to body-frame components as $\omega'= \mathbf{M}^\intercal \omega$. Why is it $\mathbf{M}^\intercal$ for angular velocity and not $\mathbf{M}$?
Is this somehow inferred from the prior steps?