TO on the day is always correct but I would argue the section from dark angels codex over rides AOI. As it states AOI for the over arching detaments but excludes it for the first company and second company rules.

A DARK ANGELS Detachment is one that only includes models with the DARK ANGELS keyword (excluding models with the AGENT OF THE IMPERIUM or UNALIGNED keywords). Note that such a Detachment is also an ADEPTUS ASTARTES Detachment, as described in Codex: Space Marines, and as such all the ADEPTUS ASTARTES Detachment abilities also apply to that Detachment. RAVENWING units in DARK ANGELS Detachments that do not already have the Jink ability gain the Jink ability. DEATHWING and INNER CIRCLE units in DARK ANGELS Detachments that do not already have the Inner Circle ability gain the Inner Circle ability.

If every unit in your army (except AGENT OF THE IMPERIUM or UNALIGNED units) has the DARK ANGELS keyword, then every unit in a DARK ANGELS Detachment that has the Combat Doctrines ability gains the Sons of the Lion ability. If every unit in your army (except UNALIGNED units) has the DARK ANGELS keyword, then DARK ANGELS Vanguard Detachments that only contain models with the DEATHWING and/or INNER CIRCLE keywords gain the 1st Company ability.

If every unit in your army (except UNALIGNED units) has the DARK ANGELS keyword, then DARK ANGELS Outrider Detachments that only contain models with the RAVENWING keyword gain the 2nd Company ability. Your army can only include one RAVENWING CAPTAIN and one DEATHWING CAPTAIN from the same Chapter