Beseech (ερωτω). For pray as in 1. John 5:16.

Lady (κυρια). Vocative case and in the same sense as in 2. John 1:1.

As though I wrote (ως γραφων). Common idiom ως with the participle (present active) for the alleged reason.

New (καινην). As in 1. John 2:7, which see.

We had (ειχαμεν). Imperfect active (late -α form like ειχαν in Mark 8:7) of εχω and note ειχετε with απ' αρχης in 1. John 2:7. Not literary plural, John identifying all Christians with himself in this blessing.

That we love one another (ινα αγαπωμεν αλληλους). Either a final clause after ερωτω as in John 17:15 or an object clause in apposition with εντολην, like 1. John 2:27; 1. John 3:23 and like verse 2. John 1:6.