As far as I know, studying quaternions, I found that the right hand rule was before 1871 because it was in that year that the London Mathematical Society accepted J.C. Maxwell's request to adopt the convention that a system is positively oriented if it rotates in the direction of the earth's rotation as seen from the north pole. Of course, reference was made to the right hand rule.
Reference: Chris Pritchard. "Tendril of the Hop and Tendril of the Vine: Peter Guthrie Tait and the Promotion of Quaternions, Part I". pp 33–34. DOI
It is also mentioned in W. R. Hamilton. "On QuaterniosQuaternions". 1847. Proceedings of the Royal Irish Academy, 3, 1–16.