His work contributed to the development of screw theory and kinematics.
In Kiev, Kotelnikov was professor and head of the department of pure mathematics until 1904.
He was at the Kyiv Polytechnic Institute directing the department of Theoretical Mechanics until 1924, when he moved to Moscow and took up teaching at Bauman Technical University.
One reviewer put Kotelnikov at the head of a chain of investigations of Spaces over Algebras.
Kotelnikov advanced an algebraic method of representing Euclidean motions that had been introduced by William Kingdon Clifford.
Though developed to render motions in three-dimensional space, an eight-dimensional algebra of doubled quaternions
According to Wilhelm Blaschke, it was Kotelnikov who initiated a "conversion principle" to take a dual rotation acting on elliptic space to a motion of