This problem has a long history, starting with the work of Killing and Cartan. You can read more here:
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Robert Bryant, Élie Cartan and geometric duality,
1998.
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Andrei Agrachev, Rolling balls and octonions, 2006.
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Ilka Agricola, Old and new on the exceptional group G2, 2008.
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Aroldo Kaplan, Quaternions and octonions in mechanics, 2008.
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John Baez and John Huerta, G2 and the rolling ball, 2012.
There is a lot more to say... but not now!
