Psalm 110:1 (LXX) does not differentiate between God and Christ. Both are called κύριος.
However, the Hebrew differentiates between God and Christ as two distinct lords (source):
God = Yahweh (the LORD)
Christ = adoni (My Lord)
Note that in Psalm 110:5 (Hebrew), Christ is said to be Adonai (= the LORD).Thus, the rendering of the LXX is justified by the Hebrew context per se.
This is interesting because in the days of Christ and the apostles (1st century A.D./ C.E.), the Septuagint usage is prevalent (source). The NT itself is written in (koine) Greek (source).
Psalm 110:3 is quoted in the Matt 22:44, Mark 12:36, Luke 20:42, Acts 2:34.
28 And one of the scribes came up and heard them disputing with one
another, and seeing that he answered them well, asked him, “Which
commandment is the most important of all?” 29 Jesus answered, “The
most important is, ‘Hear, O Israel: The Lord our God, the Lord is
one. 30 And you shall love the Lord your God with all your heart and
with all your soul and with all your mind and with all your strength.’
31 The second is this: ‘You shall love your neighbor as yourself.’
There is no other commandment greater than these.” 32 And the scribe
said to him, “You are right, Teacher. You have truly said that he is
one, and there is no other besides him. 33 And to love him with all
the heart and with all the understanding and with all the strength,
and to love one's neighbor as oneself, is much more than all whole
burnt offerings and sacrifices.” 34 And when Jesus saw that he
answered wisely, he said to him, “You are not far from the kingdom of
God.” And after that no one dared to ask him any more questions.35 And
as Jesus taught in the temple, he said, “How can the scribes say
that the Christ is the son of David? 36 David himself, in the Holy
“‘The Lord said to my Lord, “Sit at my right hand,
until I put your enemies under your feet.”’ 37 David himself calls him Lord. So how is he his son?” And the great throng heard
Mark 12:28-37 (ESV)
The son of David = The Messiah (Mark 12:35)
The Messiah = David's Lord (Mark 12:36)
The Lord = [only] one (Mark 12:29)
How can the scribes say that the Christ is the son of David?
Jesus is 'the Son of God in power according to the spirit of holiness' (Romans 1:4). Jesus is 'God over all' (Romans 9:5). Jesus himself is 'the root of David' (Revelation 22:16).
David himself calls him Lord. So how is he his son?
Jesus 'was descended from David according to the flesh' (Romans 1:3).
Jesus' twofold question reveals that he is identifying himself as Lord the same way God is.
In 1 Corinthians 8:6, Paul explicitly called Jesus Christ by the name 'one Lord'(Greek: eis kurios).
In first century Second-Temple [hellenistic] Judaism, only God is known as the 'one Lord' (Greek: eis kurios) as read in the Shema from the Septuagint but Christians ascribed this same Lordship to Jesus their Messiah which shows devotion to Jesus as God (source).