= Solution
First make the units explicit. Use <angular momentum operators> in units of $\hbar$, so their squared <eigenvalues> are $J(J+1)$ rather than $\hbar^2J(J+1)$; the coefficient $A$ then has units of <energy>. The operator identity
$$
\mathbf J^2=(\mathbf J_1+\mathbf J_2)^2
=\mathbf J_1^2+\mathbf J_2^2+2\mathbf J_1\cdot\mathbf J_2
$$
gives, at fixed $J_1,J_2$,
$$
E_J=\frac A2[J(J+1)-J_1(J_1+1)-J_2(J_2+1)].
$$
Subtracting the expression for the adjacent allowed value $J-1$ leaves
$$
\boxed{E_J-E_{J-1}=\frac A2[J(J+1)-(J-1)J]=AJ.}
$$
This is the <Landé interval rule>. Its content is that consecutive gaps grow in proportion to the higher total-angular-momentum quantum number, with the same coefficient within an isolated coupled multiplet. A negative $A$ reverses the ordering of the levels, while positive observed transition <frequencies> alone measure gap magnitudes.
If the source symbols instead denote physical angular momenta with units of action, the <Hamiltonian> written literally as $A\mathbf J_1\cdot\mathbf J_2$ gives the gap $A\hbar^2J$. Recovering the printed form then requires absorbing $\hbar^2$ into the coefficient or writing $H=(A/\hbar^2)\mathbf J_1\cdot\mathbf J_2$. The convention is essential to dimensional consistency rather than an extra dynamical assumption.
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