Yes the sabbath IS part of the 10 Commandments and Jesus was teaching about the sabbath and explained it further;
Mark 2:27-28: “The Sabbath was made for man, and not man for the Sabbath. Therefore the Son of Man is also Lord of the Sabbath.”
This indicates that the Sabbath is intended for human benefit and not a rigid rule like the pharisees were teaching . They did not understand of forgot the core of the Law, Love and Mercy and Compassion.
Matthew 12:10-14: Jesus heals a man on the Sabbath and argues that it is lawful to do good on the Sabbath. This stress the idea that the Sabbath should not be about legalistic restrictions but about compassion and mercy.
10And behold, there was a man who had a withered hand. And they asked Him, saying, “Is it lawful to heal on the Sabbath?”—that they might accuse Him.
11Then He said to them, “What man is there among you who has one sheep, and if it falls into a pit on the Sabbath, will not lay hold of it and lift
it out?
12Of how much more value then is a man than a sheep? Therefore it is lawful to do good on the Sabbath.”
13Then He said to the man, “Stretch out your hand.” And he stretched
it out, and it was restored as whole as the other.
14Then the Pharisees went out and plotted against Him, how they might destroy Him.
Jesus demonstrated the spirit of rest and the importance of genuine worship over mere ritual observance. He focused on the heart of the law rather than strict adherence to the rules.
Luke 13:15-16 Jesus challenges the Pharisees regarding healing on the Sabbath:
“The Lord then answered him and said, ‘Hypocrite! Does not each one of you on the Sabbath loose his ox or donkey from the stall and lead it away to water it? So ought not this woman, being a daughter of Abraham, whom Satan has bound—think of it—for eighteen years, be loosed from this bond on the Sabbath?’”
This shows Jesus advocating for mercy and the well-being of others on the Sabbath.
Jesus was often teaching the sabbath and how to keep it the proper way.