I am now starting to see already why this research matters. Professor Albert Pietersma is the lead translator of the NETS New English Translation of the Septuagint. This is a major scholarly work combining expertise across the Septuagint field, including Larry Perkins, whose paper on Exodus we have already discussed on this blog.
Pietersma includes in the introduction to the NETS translation of the Psalter the following note (p. 546):
Since the Greek Psalter provides no evidence that the translator made any serious attempt at distinguishing between the divine names Yahweh, including the short form "Yah", and Adonai, I have in accordance with NETS policy rendered all occurrences of kyrios, when representing either, by "Lord".
The decision is a difficult one, because even if he is right about this (no serious distinguishing going on), then you still have to work around the Adonai plus Yahweh problem with increased difficulty if both are translated by "Lord" (see Psalm 68:20: Our God is a god to save, and to the Lord Lord belong the escape routes of death).
But I don't think he is quite right about that assumption of no difference, and here're two reasons why.
1. In contrast to the article treatment in Yahweh translations in Psalms (82% are rendered anarthrous in nominative and genitive forms), when Adonai is translated into Greek only half of the translations are anarthrous (53%, that is 9 out of 17 times with respect to genitive and nominative cases - note however that one of these anarthrous instances, Ps 16:2, could almost certainly never have been confused with Yahweh in that it translates a possessive, "my Yahweh" being unheard of). This partial similarity between Yahweh and Adonai translation policy may represent a vague gesture at the sanctity of the Yahweh solution, but nonetheless, a significant difference remains (from an albeit small sampling).
2. Less significantly, but most intriguingly, is a translation of Psalm 130:6, my soul waits for the Lord. Here the Greek reads: ἤλπισεν ἡ ψυχή μου ἐπὶ τὸν κύριον. Since we are focussing on the more meaningful cases of nominative and genitive, I almost missed this accusative construction, but it rang a bell. As it turns out, "ἐπὶ" before κύριον when translating "to Yahweh" systematically removes the accusative article τὸν: Ps 4:5, Ps 21:7, Ps 22:8, Ps 31:24, Ps 32:10 and 11, Ps 37:3, Ps 40:3, Ps 55:22. Not so in Psalm 130:6 translating Adonai.
So to conclude, I can't help but wonder if Pietersma has considered these two important pieces of evidence when he dismisses the possibility of distinguishing efforts on the part of the Psalter translator. Only a more thorough investigation of the benchmark for anarthrous renderings, the Pentateuch, and the Adonai goldmine of Ezekiel will provide us with more evidence. If it can be shown that the "NETS policy" mi, in fact, be misrepresenting the translation practice of the Greek translators, then that should filter down to less clear-cut cases such as the Psalms.