It seems the confusion is coming from thinking that in the body $\mathbf{V}$ equals $\dot{\mathbf{Q}}$. This is however not true. The position vector $\mathbf{Q}$ of a point that rotates with the body is constant. (Arnol'd allows more general motions where a point $\mathbf{Q}$ moves in a rotating coordinate system $\mathbf{K}\,$ but let's stick to the case of constant $\mathbf{Q}\,.$)
It is true that in space $\mathbf{v}=\dot{\mathbf{q}}=[\omega,\mathbf{q}]$ holds, and in the body $$ \mathbf{V}=[\Omega,\mathbf{Q}]\,. 499
$$ (This is Arnol'd's notation for the cross product). But it is not true that the "velocity in the body" $\mathbf{V}$ equals $\dot{\mathbf{Q}}$ (the derivative of a constant is zero).
When $B$ is the time dependent rotation matrix around the $z$-axis,
\begin{eqnarray}\label{eZRot} B=\left(\begin{array}{ccc}\cos(\alpha\,t) & -\sin(\alpha\,t) & 0 \ \sin(\alpha\,t) & \cos(\alpha\,t) & 0\ 0 & 0 & 1 \end{array}\right)\,, \end{eqnarray} and $\mathbf{Q}$ is, say, the vector $$ \mathbf{Q}=\left(\begin{array}{c}1\0\0\end{array}\right) $$ then $$ \mathbf{q}=\left(\begin{array}{c}\cos(\alpha\,t)\\sin(\alpha\,t)\0\end{array}\right)\,,~~ \mathbf{v}=\dot{\mathbf{q}}=\alpha\left(\begin{array}{c}-\sin(\alpha\,t)\\cos(\alpha\,t)\0\end{array}\right)\,,~~ \mathbf{V}=B^\top\mathbf{v}=\left(\begin{array}{c}0\\alpha\0\end{array}\right)\,, $$ and $$ \omega=\Omega=\left(\begin{array}{c}0\ 0\ \alpha\end{array}\right)\,. $$ The equations $\mathbf{V}=[\Omega,\mathbf{Q}]$ and $\mathbf{v}=[\omega,\mathbf{q}]$ are easily verified.
Denote by $\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z$ the cartesian basis vectors fixed in space and by $\mathbf{E}_x,\mathbf{E}_y,\mathbf{E}_z$ their counterparts that are fixed in (i.e. rotating with) the body. Then clearly, $\mathbf{q},\mathbf{v},\omega$ is a system of orthogonal vectors that rotates in space around the axis $\omega=\alpha\mathbf{e}_z$, and, $$ \mathbf{Q}=\mathbf{E}_x,\quad\mathbf{V}=\alpha\mathbf{E}_y,\quad\Omega=\alpha \mathbf{E}_z $$ is a system of orthogonal vectors that is fixed in the body. 804