This all makes sense, but what about those off-diagonal terms in ? Consider the vector angular momentum (§B.4.14):
We see that the off-diagonal terms correspond to a coupling of rotation about with rotation about . That is, there is a component of moment-of-inertia that is contributed (or subtracted, as we saw above for ) when both and are nonzero. These cross-terms can be eliminated by diagonalizing the matrix [452],^{B.25}as discussed further in the next section.