Show that any proper homogeneous Lorentz transformation may be expressed as the product of a boost times a rotation

Question

I am trying to read Weinberg's book Gravitation and Cosmology. In which he derives the Lorentz transformation matrix for boost along arbitrary direction, (equations 2.1.20 and 2.1.21):
$$Lambda^i_{,,j}=delta_{ij}+v_i v_jfrac{gamma-1}{mathbf{vcdot v}}$$
$$Lambda^0_{,,j}=gamma v_j$$
Immediately after that there is a statement, "It can be easily seen that any proper homogeneous Lorentz transformation may be expressed as the product of a boost times a rotation".
How to show that mathematically? It'd be better if someone answers using similar notations as used by Weinberg.

DanielC · Accepted Answer

It is not so readily seen, to be honest. It goes in the literature by the name "polar decomposition".
The shortest argument is the one by H. Urbantke "Elementary Proof of Moretti’s Polar Decomposition Theorem for Lorentz Transformations" (here), which is a not so strightforward simplification of prof. Valter Moretti's argument here. You need Sexl & Urbantke's famous book to follow Urbantke.

Show that any proper homogeneous Lorentz transformation may be expressed as the product of a boost times a rotation

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