= Solution
In <LS coupling>, add all orbital and <spin> <angular momenta> separately:
$$
\mathbf L=\sum_a\mathbf l_a,\qquad
\mathbf S=\sum_a\mathbf s_a,\qquad
\mathbf J=\mathbf L+\mathbf S.
$$
The <quantum numbers> specify
$$
\begin{aligned}
L^2&\longrightarrow\hbar^2L(L+1),&L_z&\longrightarrow\hbar M_L,\\
S^2&\longrightarrow\hbar^2S(S+1),&S_z&\longrightarrow\hbar M_S,\\
J^2&\longrightarrow\hbar^2J(J+1).
\end{aligned}
$$
Thus $M_L=-L,\ldots,L$ and $M_S=-S,\ldots,S$. The <addition of angular momentum> rule gives
$$
\boxed{J=|L-S|,|L-S|+1,\ldots,L+S.}
$$
In a product basis with individual projection labels,
$$
\boxed{M_L=\sum_a m_{l,a},\qquad M_S=\sum_a m_{s,a}.}
$$
The magnitudes $L$ and $S$ are not generally the arithmetic sums of the individual magnitudes; they result from angular-momentum coupling. The sum rules for projections remain valid for every product component of a coupled <angular momentum eigenstate>.
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