Higher dimensional relation between angular momentum, moment of intertia and angular velocity

It turns out the answer is rather simpler in higher dimensions. It gets complicated when specializing to 3-dimensions.

If a rigid body has angular velocity $\omega_{ij}$ then given the location of a point $r_i$ its velocity is given by

$$ v_i = -\omega_{ij} r_j $$

and its angular momentum is given by

$$ L_{ij} =\sum m(r) ( -r_i v_j +r_j v_i )\\ = \sum m(r)( r_i r_m \omega_{mj} -r_j r_m \omega_{mi}) \\ = 2 [\sum m(r) r_i r_m] \omega_{mj} $$ where in the last expression one must remember the anti-symmetry of $L_{ij}$.

Specializing to 3D we get $$ \tilde L_k = \frac{1}{2} \epsilon_{kij} L_{ij}\\ =\epsilon_{mjt}\epsilon_{ijk} [\sum m(r) r_i r_m] \tilde \omega_t \\ = -\delta^{tm}_{ik} [\sum m(r) r_i r_m ] \tilde \omega_t \\ = [\sum m(r) (-r_k r_i +r^2 \delta_{ki})] \tilde \omega_i \\ = I_{ki} \tilde \omega_i $$

Thus we see that in arbitrary dimensions the relation between angular momenta, moment of inertia and angular velocity is given by

$$ L_{ij} = 2 \tilde I_{im} \omega_{mj} $$

where $\tilde I_{ij} = \sum m(r) r_i r_j$.