# Relationship of SU(2) & SU(2) to SO(4)?

While almost all accessible references indicate/demonstrate that group SO(4) = SU(2)⊗SU(2), I've come across two references that state the relationship as SO(4) = SU(2)⊕SU(2).

Is the latter equation referring to the Lie algebra? Or is it a typo?

Clarification appreciated.

• Could you give a reference saying that $SO(4)=SU(2)\otimes SU(2)$? What is true is that $$(SU(2)\times SU(2))/{\Bbb Z_2}\cong SO(4).$$ The $\oplus$ is for vector spaces, i.e., for Lie algebras. – Dietrich Burde Mar 1 at 12:21
• @DietrichBurde Sure: arxiv.org/pdf/quant-ph/0608186v2.pdf . And if the + is for the Lie Algebras, that answers my question. Many thanks – iSeeker Mar 1 at 12:24
• Is this the tensor product considered as representations? For the other isomorphism see here. – Dietrich Burde Mar 1 at 12:42
• @DietrichBurde It could also just be physicists using different notation (I have seen them use $\otimes$ for direct product before at least). – Tobias Kildetoft Mar 1 at 12:58
• @TobiasKildetoft I see. But then, how to deal with the non-trivial kernel $\Bbb Z_2$? – Dietrich Burde Mar 1 at 13:00