Finite rotations and infinitesimal rotations

[wrobel]

With Lie-group theory that's very easy to understand.

Oh! Now I see what you call Lie group theory. This is not yet Lie group theory this is just some trivial argument. Lie group theory is a serious mathematical branch. By the way, your formulas are too cumbersome. Particularly there is no need to refer on coordinate representations every time. Everything is much simpler.

Let $u$ be any vector frozen into a rigid body. The rigid body moves so that $u=u(t)$. Define a linear operator $$A(t)u(0)=u(t).\qquad (!)$$ It is a unitary operator it's clear:
$$AA^*=I,\qquad (*)$$
Actually, formulas (*) and (!) are the definition of a rigid body.

Differentiate formula (!):
$$\dot u(t)=\dot A(t)u(0)=\dot A(t)A^{-1}(t)u(t)=\dot A(t)A^*(t)u(t).\qquad (**)$$
The operator $\Omega=\dot AA^*$ is skew-symmetric: $\Omega^*=-\Omega$. Indeed, just take $\frac{d}{dt}$ from both sides of (*).

And finally, any skew-symmetric operator in $\mathbb{R}^3$ can unequally be presented with the help of (pseudo) vector such that $\Omega x=\omega\times x,$ and correspondingly formula (**) takes the form $\dot u(t)=\omega\times u(t)$

 

[vela]

No, it's very simple!

Simple for you, but probably not to the OP! Personally, I think talk of Lie groups, Lie algebra, and the like is too advanced for an I-level thread.