Zhao Su was a fifth-generation descendant of Shu Dai 叔帶, who was himself a descendant of Zao Fu 造父, who had lived during the reign of King You 周幽王, the last ruler of the Western Zhou period 西周 (11th cent.-770 BCE), and had decided to become a vassal of Jin rather than of the licentious King You.
Shu Dai was allowed by the duke of Jin to bear the family name Zhao, which was borrowed from the name of the fief of Shu Dai, Zhaocheng 趙城 or Zhaoyuan 趙原. In 660, Duke Xian of Jin 晉獻公 (r. 677-651) entrusted Zhao Su with the attack on the three minor fiefs of Huo 霍, Wei 魏 and Geng 耿. As a reward, Zhao Su was presented the territory of Geng. This was the formal beginning of the family Zhao being vassals to the state of Jin.