The optimization process and recommendations:

1. Avoid Selecting Unnecessary Columns (query line: 2): Avoid selecting all columns with the '*' wildcard, unless you intend to use them all. Selecting redundant columns may result in unnecessary performance degradation.
2. Avoid Subselect When Selecting MAX/MIN Per Group (query line: 7): Constant subquery results are usually not cached by the database, especially in non-recent database versions. Therefore, a constant subquery in a WHERE clause will be fully evaluated for every row the WHERE clause will examine, which can significantly impact query performance. Use the method mentioned in the example instead.
3. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

CREATE INDEX event_idx_id_state_sub_id ON "event" ("id","state","sub_id");
CREATE INDEX event_idx_sub_id_state ON "event" ("sub_id","state");

The optimized query:

SELECT
        * 
    FROM
        event AS event1 
    LEFT JOIN
        event AS event2 
            ON (
                event1.sub_id = event2.sub_id 
                AND event2.state = 'PENDING' 
                AND event2.il_flag = event2.true
            ) 
            AND (
                event1.id < event2.id
            ) 
    WHERE
        (
            1 = 1 
            AND NOT EXISTS (
                SELECT
                    se2.id 
                FROM
                    event se2 
                WHERE
                    se1.sub_id = se2.sub_id 
                    AND se2.state IN (
                        'ACCEPTED', 'FAILED', 'INPROCESS'
                    )
            )
        ) 
        AND (
            event2.id IS NULL
        )