I want to compute the angles between the principal axis (i, j, k) and the atomic axis (x, y, z). So I'm using coor inertia to get the principal axis but I'm a bit confused as to what the secondary and tertiary axes describe. Could someone please elaborate?

Computing the inertial axes is a 3D eigenvalue problem, with the principal axis corresponding to the smallest eigenvalue, i.e. the axis with least inertia wrt. rotation around the axis.

The secondary and tertiary axes are orthogonal to the primary axis, with successively larger inertia (eigenvalues), as more mass is rotated.

One thing to note is that the direction of the principal axis is arbitrary, in terms of the sign of the vector components. For example, the principal inertial axis of a helix could point in either direction.

I see. So the components of the principal, secondary and tertiary axes are essentially the cosine of the angles of rotation.

The inertial axes are unit vectors, so yes the components correspond to cosines wrt. to the lab frame axes.