From Dr. Michael Horton’s systematic, The Christian Faith, page 578
Referring to Psalm 51 Dr. Horton writes:
First, we recognize that David is not ashamed of his behavior but guilty. Second, although he has sinned cruelly against Bathsheba and plotted the death of her husband, he recognizes that his sin is first and foremost against God. Repentance is not only remorse for having wronged our neighbor, but is a recognition that God is the most offended party. Third, David does not try to atone for his sins or pacify God’s just anger by his remorse. David confesses that before God’s throne he is condemned, and he does not try to justify himself. Fourth, David acknowledges not only his sinful actions but his sinful condition from the hour of conception. Repentance pertains not only to certain sins; pagans can be remorseful for their immoderate behavior. Rather, it is the revulsion of the whole soul toward its alliance with sin and death.