I have a rotation stored in a quaternion $\textbf{q}$ and an inertia tensor of a rigidbody $\mathit{I_b}$ ($b$ for body) stored in a $3\times 3$ matrix.
I would like to express my inertia tensor $\mathit{I_b}$ in a rotated frame $\{c\}$. luckily there is a formula utilizing a rotation matrix $\mathit{R_{bc}}$
$$\mathit{I_c} = \mathit{R^T_{bc}}\mathit{I_b}\mathit{R_{bc}}$$
What I'm wondering about is if it is possible to replace the rotation matrices with quaternions and to "combine" them directly with the inertia tensor $\mathit{I_b}$ WITHOUT first converting the quaternion to a $3\times3$ matrix.
In short order how would you multiply a quaternion or quaternion transpose by or into a $3\times3$ matrix?
perhaps something like this?
$$\mathit{I_c} = \mathit{q^T}\mathit{I_b}\mathit{q}$$
where $\mathit{q}$ is a unit quaternion representing rotation & $\mathit{q^T}$ is said quaternion's transpose.