TITLE: Recovering the two $SU(2)$ matrices from $SO(4)$ matrix
QUESTION [22 upvotes]: Since there is a $2$-$1$ homomorphism from $SU(2)\times SU(2)$ to $SO(4)$ there should be a way to recover the two $SU(2)$ matrices given an $SO(4)$ matrix.  
I believe I could set this up as a system of equations using the map from above and solve for the coefficients of the $SU(2)$ matrices.  However, I wonder if anyone has already done this or can point me to the formulas?

REPLY [4 votes]: In terms of SU(2) = unit quaternions, the answer can be found in the sections "Isoclinic decomposition" and "Relation to quaternions" of the Wikipedia article about SO(4).
(This is equivalent to Robin Chapman's answer, I presume, but the formulas are a bit more explicit.)