That’s exactly what I mean, there are no inverse trig that needs to be involved anywhere, see the matrix section in the article. It is trivial to recognize that vector inputs are pretty much the same space the k coefficients and mentally treat cos and sin as a shorthand symbol in terms of vector products.

And in any case inverse trig functions are just components of logarithms of quaternions, and trig is components of exponentiating half-axis-angle into quaternions.