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The algebra of the Lorentz group $SO(3, 1)$ can be represented as direct product of $SU(2)$ or $SO(3)$ algebras. How to understand this statement?

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It means that they're subgroups (or can be viewed as subgroups in a natural way) and any element in the bigger group can be uniquely expressed as the product of an element in $SU(2)$ and an element of $SO(3)$. –  zibadawa timmy Nov 17 '13 at 17:23
    
@zibadawatimmy . The Lorentz group has two subgroups, which have independent algebras of SU(2) (or SO(3)) group. So your statement means that all of matrices of the lorentz group matrix representation can be expressed as the product of two elements of SU(2) group? –  John Taylor Nov 17 '13 at 17:29
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I think I missed the "algebras" and that it was an "or" instead of an "and" the first time around, but yes, give or take that it's weird (to me) to just call the parent space a group but then split it into algebras. I do not know the exact such algebras, or what their physical interpretation is, though. –  zibadawa timmy Nov 17 '13 at 17:44

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