How to prove SO(1,3)=SU(2)+SU(2)

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SUMMARY

The discussion centers on the relationship between the Lie Algebra of SO(1,3) and the direct sum of SU(2) and SU(2). It is established that SO(1,3) does not decompose into SU(2)+SU(2) but rather that the complexified Lie algebras of these groups are isomorphic. To demonstrate this, one must exhibit a basis in one Lie algebra that satisfies the commutation relations of the other, necessitating the use of complex coefficients for proper representation.

PREREQUISITES
  • Understanding of Lie Algebras
  • Familiarity with SO(1,3) and SU(2) group structures
  • Knowledge of complexification in algebraic structures
  • Comprehension of commutation relations in Lie Algebras
NEXT STEPS
  • Study the properties of Lie Algebras and their representations
  • Learn about complexification in algebraic contexts
  • Research the commutation relations specific to SO(1,3) and SU(2)
  • Explore the implications of isomorphism in Lie Algebras
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The discussion is beneficial for mathematicians, theoretical physicists, and students studying advanced algebraic structures, particularly those interested in the interplay between different Lie groups and their algebras.

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The Lie Algebra of SO(1,3) is similar with that of SU(2)+SU(2) or of SO(3)+SO(3). But how do we know SO(1,3) really decomposite to SU(2)+SU(2)?
 
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Your statement is false, hence cannot be proved. Only the conplexified Lie algebras of the groups you write down are isomorphic.

To prove this you exhibit in one of the Lie algebras a basis satisfying the commutation rules of the other. Since complex coefficients appear, complexification is necessary.
 
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