  In this paper we investigate the Besov spaces on compact Lie groups in a
subelliptic 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 spaces
associated to the Laplacian on the group are proved. We link the description of
subelliptic Sobolev spaces with the matrix-valued quantisation procedure of
pseudo-differential operators in order to provide subelliptic Sobolev and Besov
estimates for operators in the H\"ormander classes. Interpolation properties
between Besov spaces and Triebel-Lizorkin spaces are also investigated.
