Using the fact that the angular velocity is a vector as infinitesimal rotation commute.

Then the relative angular velocity of body 1 w.r.t 2 is given by

w₁₂ = w₁ -w₂     (as for relative linear velocity)

The relative acceleration of 1 w.r.t 2 is

(dw₁ / dt)_S'

Where S' is a frame rotating with the second body and S is a space fixed frame with origin coinciding with the point of intersection of the two axes,

but (dw₁ / dt)_S = (dw₁ / dt)_S' + w₁ X w₂

Since S' rotates with angular velocity w₂. However (dw₁ / dt)_S = 0 as the first body rotates with constant angular velocity in space, thus

Beta₁₂ = w₁ X w₂

Note that for any vector b, the relation in space forced frame (k) and a frame (k') rotating with angular velocity w is

db/dt I_K = db/dt I_{K '} + w X b