Decomposition of restricted Lorentz transformation matrix

Question

Please can someone give the proof of all restricted Lorentz transformation matrix can uniquely decomposed into product of pure rotation and pure boost? Or at least give some hint how to proceed to prove it?

Rob Tan · Answer

If I understood your question, what you should do is write the Lorentz transformation $Lambda$ as the exponential of a matrix $e^{mathbb{L}}$, as is the form of the transformation of a Lie group and from the generic condition of all Lorentz transformation $Lambda^Tmathbb{G}Lambda=mathbb{G}$ you should arrive to prove that $(mathbb{G}mathbb{L})^T=-mathbb{G}mathbb{L}$, where $mathbb{G}doteq(G_{alphabeta})$ is the metric tensor (while $mathbb{L}$ matrix has components of the form ${{mathbb{L}}^alpha}_beta$). You should conclude that the matrix has this form
$$
mathbb{L}
=
begin{pmatrix}
0&{mathbb{L}^0}_1&{mathbb{L}^0}_2&{mathbb{L}^0}_3

{mathbb{L}^0}_1&0&{mathbb{L}^1}_2&{mathbb{L}^1}_3

{mathbb{L}^0}_2&-{mathbb{L}^1}_2&0&{mathbb{L}^2}_3

{mathbb{L}^0}_3&-{mathbb{L}^1}_3&-{mathbb{L}^2}_3&0
end{pmatrix}
$$
and that's it. The matrices associated to ${mathbb{L}^0}_1,{mathbb{L}^0}_2,{mathbb{L}^0}_3$ parameters are the generators of the boosts, the others represent the generators of rotations.
Edit:
begin{gather*}
boldsymbol{beta}
doteq
left(
{mathbb{L}^0}_1,
{mathbb{L}^0}_2,
{mathbb{L}^0}_3
right),
,
boldsymbol{alpha}
doteq
left(
-{mathbb{L}^1}_2,
{mathbb{L}^1}_3,
-{mathbb{L}^2}_3
right)

mathbb{K}_1
doteq
	begin{pmatrix}
	0&1&0&0
	
	1&0&0&0
	
	0&0&0&0
	
	0&0&0&0
	end{pmatrix},
,
mathbb{K}_2
doteq
	begin{pmatrix}
	0&0&1&0
	
	0&0&0&0
	
	1&0&0&0
	
	0&0&0&0
	end{pmatrix},
,
mathbb{K}_3
doteq
	begin{pmatrix}
	0&0&0&1
	
	0&0&0&0
	
	0&0&0&0
	
	1&0&0&0
	end{pmatrix}

mathbb{J}_1
doteq
	begin{pmatrix}
	0&0&0&0
	
	0&0&0&0
	
	0&0&0&-1
	
	0&0&1&0
	end{pmatrix},
,
mathbb{J}_2
doteq
	begin{pmatrix}
	0&0&0&0
	
	0&0&0&1
	
	0&0&0&0
	
	0&-1&0&0
	end{pmatrix},
,
mathbb{J}_3
doteq
	begin{pmatrix}
	0&0&0&0
	
	0&0&-1&0
	
	0&1&0&0
	
	0&0&0&0
	end{pmatrix}
%end{drcases}

mathbb{L}
equiv
beta^imathbb{K}_i
+alpha^imathbb{J}_i
end{gather*}
As you can see $boldsymbol{alpha}$ are the parameters of rotation, $boldsymbol{beta}$ of boost. The following commutation relations hold
begin{gather*}
begin{cases}
	left[
	mathbb{K}_i,
	mathbb{K}_j
	right]
	=
	-{epsilon_{ij}}^k
	mathbb{J}_k
	
	left[
	mathbb{J}_i,
	mathbb{J}_j
	right]
	=
	{epsilon_{ij}}^k
	mathbb{J}_k
	
	left[
	mathbb{J}_i,
	mathbb{K}_j
	right]
	=
	{epsilon_{ij}}^k
	mathbb{K}_k
	end{cases}
end{gather*}

Decomposition of restricted Lorentz transformation matrix

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