Applying torque to rigid body on a fixed axel?

Suppose I have a torque $tau in mathbb{R}^3$ on a rigid body, about its center of gravity, where the direction of $tau$ is the axis of the torque and the magnitude of $tau$ is the strength of the torque.

Let’s suppose the rigid body is on some infinitely strong fixed axel, through its center of gravity, with unit vector axis $alpha in mathbb{R}^3, |alpha| = 1$. Notice that $tau$ and $alpha$ may be in different directions.

The torque $tau$ is therefore equivalent to some torque $kalpha$ where $k in mathbb{R}$.

If $tau$ and $alpha$ are perpendicular ($tau cdot alpha = 0$) then $k$ is zero because the torque would be entirely absorbed by the axel.

If $tau$ and $alpha$ are collinear, then there exists some $c in mathbb{R}$ such that $calpha = tau$. In such a case none of the torque is absorbed by the axel, and $k = c$

For the remaining cases what is $k$ in terms of $tau$ and $alpha$ ?