In 1235, when he was first mentioned by contemporary records, Matthew had already begun his political career as a well-known member of junior king Béla's court, therefore it was not necessary to distinguish him from the other lords by specifying the name of his father.
[4] Their son, Stephen Sternberg (or "the Bohemian") later inherited the Csák dominion because of the absence of a direct adult male descendant after the death of Matthew III in 1321.
Béla's main purpose was to restore the royal power that had weakened during his father's rule; e.g., he ordered the burning of his advisors' seats, because he wanted to force them to stand in the presence of the king.
Furthermore, Béla dismissed his father's former loyal followers and advisors (during this time, for example, former palatines Denis, son of Ampud was blinded and Julius Kán was imprisoned).
[7] Following the disastrous Battle of Mohi (11 April 1241), Béla IV fled the scene to Spalato (today: Split, Croatia), where Matthew I was also a member of the king's men.
According to a royal charter issued in 1249, Béla IV formerly also donated Racsic (Slovak: Račice; today part of Nitrica) to Matthew, who was already dead by that time.
Consequently, he did not initiate his family through the establishment of oligarchical power with land acquisitions, he just marked its direction (from the ancient estates of the genus in Fejér County to northwest).