Is it permissible to purchase electrical appliances by installments through a brother who is a government employee, from an interest-based bank, knowing that the installment price does not differ from the cash price and there are no interests?

If the bank will pay the agreed price for the commodity and collect it from your brother in installments without any increase or interest, then there is no harm in the transaction; because it falls under the category of a benevolent loan (Qard Hasan). It is stipulated that the contract must not contain any corrupt condition, such as a late payment penalty. Your brother should be advised to close his account in the usurious bank and open an account in one of the Islamic banks, if available. It is not permissible to open an account in a usurious bank if other options exist, unless no other options are available and he is in need of it, in which case opening a current account is permissible.