Angular-momentum vectors of the ξ Tau sub-systems (Aa+Ab, B, C) and their long-term evolution. The coordinates were rotated to the Laplace reference frame, with the z-axis along the total angular momentum, L_sum, of the whole ξ Tau system (in absence of external torques L_sum is constant). The individual contributions were obtained as ${\mathit{L}}_{\mathrm{A}}=m_{\mathrm{A}}\prime {\mathit{r}}_{\mathrm{A}}\times {\dot{\mathit{r}}}_{\mathrm{A}}$, where m_A′ denotes the reduced mass, r_A and ṙ_A the Jacobi coordinate and velocity of the Aa+Ab binary (and similarly for the B and C parts, L_B and L_C). The (2+1)+1 hierarchy of the system implies that (i) L_C and L_AB ≡ L_A + L_B regularly precess about L_sum with a period of ≃85000 y (see colours showing the time in the course of simulation), preserving their mutual angle J₂ ≃ 71°; (ii) L_A and L_B regularly precess about L_AB with a period of ≃7000 d, preserving their mutual angle of J₁ ≃ 0.5°. Since L_C holds the largest share of L_sum, its angular distance from L_sum is only 23°, while that of L_AB is more than 47°. As a result, the observed inclinations with respect to the sky plane, i₁ and i₂, exhibit large variations over millennia.