As explained in the introduction section of these notes, the "elder" was evidently the Apostle John , the "chosen lady" a local church, and her "children" the believers in that church. [source][source][source]
"It may well be that the address is deliberately unidentifiable. The letter was written at a time when persecution was a real possibility. If the letter were to fall into the wrong hands, there might well be trouble. And it may well be that the letter is addressed in such a way that to the insider its destination is quite clear, while to the outsider it would look like a personal letter from one friend to another. The address may in fact be a skilful attempt to baffle any hostile person into whose hands the letter might come; and, if that is Song of Solomon , our difficulty in identifying the person or Church to whom the letter is addressed is nothing other than a tribute to the skill of John." [1][source]
The church was "chosen" in that it consisted of elect individuals: Christians. [source][source][source]
"We are hardly to think here of an elder in the sense which the word presbyteros usually bears in Christian contexts in the New Testament, that Isaiah , one who discharges the ministry of eldership in a local church.... The word appears in another specialized sense in second-century Christian literature, of church leaders in the generation after the apostles, particularly those who were disciples of apostles or of "apostolic men," and were therefore guarantors of the "tradition" which they received from the apostles and delivered in turn to their own followers." [2][source]
John loved this church and so did other Christians who knew about it. The basis of this love was the truth the Christians there believed in common with one another. This "truth" refers to God"s revelation in Scripture. The importance of this truth is clear from the fact that John referred to it three times in these two verses. [source][source][source]
"The Truth makes true love possible." [3][source]