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In this paper we investigate the Besov spaces on compact Lie groups in asubelliptic setting, that is, associated with a family of vector fields,satisfying the H\"ormander condition, and their corresponding sub-Laplacian.Embedding properties between subelliptic Besov spaces and Besov spacesassociated to the Laplacian on the group are proved. We link the description ofsubelliptic Sobolev spaces with the matrix-valued quantisation procedure ofpseudo-differential operators in order to provide subelliptic Sobolev and Besovestimates for operators in the H\"ormander classes. Interpolation propertiesbetween Besov spaces and Triebel-Lizorkin spaces are also investigated.

Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups