= Solution
Set $\omega_1=\omega_2=\omega$. Inverting the <creation and annihilation operators> gives
$$
X_j=\sqrt{\frac{\hbar}{2m\omega}}(a_{j,+}+a_{j,-}),\qquad
P_j=i\sqrt{\frac{m\hbar\omega}{2}}(a_{j,+}-a_{j,-}).
$$
Substitute these expressions in $L_3=X_1P_2-X_2P_1$. Terms with two <creation operators> or two <annihilation operators> cancel because different modes commute. The remaining <orbital angular momentum in oscillator ladder operators> is
$$
\boxed{L_3=i\hbar\left(a_{2,+}a_{1,-}-a_{1,+}a_{2,-}\right).}
$$
Equivalently it is $-i\hbar(a_{1,+}a_{2,-}-a_{2,+}a_{1,-})$, consistent with the <circular ladder operators of a two-dimensional oscillator>.
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