What is incumbent upon me to atone for the mistake of bearing responsibility for a car accident caused by my brother, knowing that I am insured against third parties and my brother is not?

Insurance of all types - except cooperative insurance - is forbidden gambling (maysir). Any contract where the contractor is either a loser or a winner is considered gambling. Therefore, whoever enters into a non-cooperative insurance contract, the contract is void. The insured has the right to reclaim what they paid without any increase, and if they took more, they must return the excess. It is not permissible to continue in it except under duress.

If you took out insurance without being compelled, then you are sinful and repentance is incumbent upon you. If you were compelled, then the sin is upon the one who compelled you.

If what your brother damaged is equal to or less than your capital paid to the insurance company, then there is nothing upon you. Otherwise, you must return any amount exceeding the capital to the insurance company.