Prince Simphiwe Zulu and Prince Vanana Zulu, royals within King Misuzulu kaZwlithini’s inner circle, had been travelling from Pretoria to KwaZulu-Natal on Thursday evening when the incident happened.
The spokesperson, Prince Africa Zulu, says the princes are not in “critical” condition and only sustained “minor” injuries.
This accident comes as the Zulu royal family is embroiled in a legal squabble over King Misuzulu’s claim to the throne.
Prince Simakade, his half-brother, wants the court to overturn President Cyril Ramaphosa’s recognition of King Misuzulu as the rightful heir and name him monarch instead.
He previously tried to stop the coronation of King Misuzulu – but his attempts were unsuccessful.
The Zulu royal family receives a household allowance of almost 80 million rand ($4.2m; £3.4m) and owns 30% of the land in Kwa-Zulu Natal province.