I found the following definition of angular velocity vector of B in A at page 49 of the book "Thomas R. Kane, Peter W. Likins, David A. Levinson - Spacecraft Dynamics - McGraw-Hill (1981)":
The problem is the notation used. If I want the angular velocity vector of B relative to A, I have to write each of the 3 terms in A reference frame ($a_1, a_2, a_3$ unit vectors of A reference frame; $b_1,b_2,b_3$ unit vectors of B reference frame:
$$ \overrightarrow{\omega}_{A\rightarrow B}\Bigr|_{A} = % % \left( \overrightarrow{\omega}_{A\rightarrow B}\Bigr|_{A} \cdot \hat{b}_1\Bigr|_{A} \right) \hat{b}_1\Bigr|_{A} + % % \left( \overrightarrow{\omega}_{A\rightarrow B}\Bigr|_{A} \cdot \hat{b}_2\Bigr|_{A} \right) \hat{b}_2\Bigr|_{A} + % % \left( \overrightarrow{\omega}_{A\rightarrow B}\Bigr|_{A} \cdot \hat{b}_3\Bigr|_{A} \right) \hat{b}_3\Bigr|_{A} = % % % \omega_{b_1,A\rightarrow B}\Bigr|_{A} \hat{b}_1\Bigr|_{A} + % % \omega_{b_2,A\rightarrow B}\Bigr|_{A} \hat{b}_2\Bigr|_{A} + % % \omega_{b_3,A\rightarrow B}\Bigr|_{A} \hat{b}_3\Bigr|_{A} % % $$
where $\omega_{b_i,A\rightarrow B}\Bigr|_{A}$ is the component along $\hat{b}_i$ of the angular velocity vector of B relative to A expressed in A. Instead $\hat{b}_i\Bigr|_{A}$ is the unit vector $\hat{b}_i$ of B expressed in A.
Is this right?
EDIT: in Sec. 1.10 the author said that A and B are 2 rigid bodies which are moving relative to each other:
Thank you in advance.