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

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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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A. Neumaier
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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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