A new method for developing signature schemes on finite noncommutative associative algebras is introduced. A signature algorithm is developed on a 4-dimensional algebra defined over the ground field GF(p). The public key element and one of the signature elements represent vectors calculated using exponentiation operations in a hidden commutative group. Decomposition of the algebra into commutative subalgebras is taken into account while designing the algorithm. The method extends the class of algebraic digital signature schemes and opens up the possibility of developing a number of practical post-quantum digital signature algorithms, the main merit of which is comparatively small size of the public key, secret key, and signature.