is and forget are the verbs. Brother is the subject
There are two clauses in this sentence. The first clause (Surely your brother is dependable. )
brother is the subject of the first clause is, is the verb of the first clause. Is a a statement of being.
The second clause is (will not forget his promises.)
brother is the understood subject of the second clause.
his is a pronoun but not the subject form which would be he.
his is the possessive form of the pronoun and can not be the subject but refers back to brother in the first clause.
will and not are adverbs that describe the verb forget.
The sentence could be rewritten for clarity
Surely your brother is dependable and he will not forget his promise