Anomalous Zeeman Effect I was reading splitting of spectral lines in magnetic field and my book says

In anomalous Zeeman Effect,
Classically, the ratio of orbital angular moment to angular momentum $|\vec{L}|$ gives us $\vec{\mu_L} =\frac{e}{2m}\vec{|L|}$  and the ratio of spin angular moment to spin momentum $|\vec{S}|$ we have $\vec{\mu_S}=\frac{2e}{2m}\vec{|S|}\;\;$  [ Anomalous Zeeman effect requires this and Quantum Mechanics confirms it.]
Because of this inequality $\vec{\mu_J} = \vec{\mu_L} + \vec{\mu_S}$ is not exactly anti-parallel to $\vec{J}$.

I understood it till now and then my book says

since $\vec{J}$ is invariant $\vec{L},\vec{S},\vec{\mu_J},\vec{\mu_S} \; $and $\; \vec{\mu_L}$ precess around $\vec{J}$.

What does this last statement mean ?
 A: For a atom in a external magnetic field $\boldsymbol{B}$, there are 3 regimes, weak (inclusive of 0), strong, and in between, which are explicitly solved by use of the Anomalous Zeeman Effect, Paschen-Back Effect, and use of Breit-Rabi formula (to not be discussed here) respectively.
Noting that these behaviour defer from the "Normal" Zeeman effect when we introduce spin into electrons of the Bohr Model. For addition of angular momentum, one should not think of literal angular momentum vectors in $\mathbb{R}^3$ space being added, but it's the most common way by which this is used to explain the various Zeeman effects, and so, we shall. Now to answer the question:

since $\vec{J}$ is invariant, $\vec{L}$, $\vec{S}$, $\vec{\mu}_J$, $\vec{\mu}_L$, $\vec{\mu}_S$ precesses around $\vec{J}$.

In the vector model, these are how all the vector quantities would look like.
 
We note that the vector addition must still satisfy the quantisation rules.
In the case of a weak field, the external field $\boldsymbol{B}$ exerts a torque $\boldsymbol{D}$ on $\boldsymbol{J}$ given by $$\boldsymbol{D} = \boldsymbol{\mu}_J \times \boldsymbol{B}$$ such that $\boldsymbol{L}$ and $\boldsymbol{S}$ do not decouple, and hence, we say that $\boldsymbol{J}$ is invariant. For a strong field, $\boldsymbol{J}$ is no longer invariant, and $\boldsymbol{L}$ and $\boldsymbol{S}$ would precess about the $\boldsymbol{B}$ field at their own rates.
For the various spin magnetic moments, we take a look at the following diagram.
 
Note that $\boldsymbol{\mu}_J$ is not parralel to $\boldsymbol{J}$ simply because $$ \boldsymbol{\mu}_J = \boldsymbol{\mu}_L + \boldsymbol{\mu}_S = -\frac{\mu_B}{\hbar}\left( \boldsymbol{L} + g_S \boldsymbol{S} \right), \quad g_S \approx 2. $$
That said, we can find the time average $\langle \boldsymbol{\mu}_J \rangle$ of $\boldsymbol{\mu}_J$ is just the projection of $\boldsymbol{\mu}_J$ onto $\boldsymbol{J}$.
This should answer the question.

A more proper treatment of these $\boldsymbol{L}$ and $\boldsymbol{S}$ in external magnetic fields and its consequence on the energy level splittings is properly explained using time-independent pertubation theory.

Figures sourced from reference: Wolfgang Demtröder's Atoms, Molecules and Photons. Chapter 5.5.5
