Geodesic
The geometry of the Lie algebra of the orthogonal group $O(\mathbb R^4)$
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 119-124

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In the $6$-dimensional space $\Lambda_2(\mathbb R^4)$ of bivectors a Lie product is introduced analogous to the standard vector product in $\mathbb R^2$. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the group of orthogonal transformations $O(\mathbb R^4)$. This isomorphism of Lie algebras is a canonical isometry of the space of antisymmetric operators in $\mathbb R^4$ onto $\Lambda_2(\mathbb R^4)$. 
@article{ZNSL_1999_261_a8,
     author = {S. E. Kozlov and M. Yu. Nikanorova},
     title = {The geometry of the {Lie} algebra of the orthogonal group $O(\mathbb R^4)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--124},
     publisher = {mathdoc},
     volume = {261},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a8/}
}
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S. E. Kozlov; M. Yu. Nikanorova. The geometry of the Lie algebra of the orthogonal group $O(\mathbb R^4)$. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 119-124. http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a8/