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2012f.00785 Fujii, Kazuyuki $SO(4)$ re-revisited. Far East J. Math. Educ. 8, No. 2, 85-99 (2012). 2012 Pushpa Publishing House, Allahabad, Uttar Pradesh, India EN H60 $SO(4)$ exponential maps magic matrices explicit expressions

http://www.pphmj.com/abstract/6637.htm

Summary: In this note, an explicit expression of $\exp A \ (A \in SO(4))$ is given in terms of the magic matrix by Makhlin.