Galatians 3:19,
Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator.
What is the relationship with angels and the law?
Gal 3:19 says: Why then was the law given? It was added because of transgressions, until the arrival of the seed to whom the promise referred. It was administered through angels by a mediator.
The final sentence of this verse is: διαταγεὶς δι’ ἀγγέλων ἐν χειρὶ μεσίτου. I would translate this (very literally): "having been organised/ordained by angels in [the] hand of a mediator."
The OP askes about the first two words quoted above in the Greek, the first of which is διατάσσω (diatassó) which occurs 16 times in the NT. It is variously translated as arrange (1), arranged (1), commanded (3), direct (1), directed (4), gave orders (2), giving instructions (1), ordained (1), ordered (1), orders (1).
BDAG gives the meaning of διατάσσω (diatassó) as:
The giving of the law at Sinai "by angels" is regularly spoken about in the Bible: Deut 33:2 (LXX), Acts 7:38, 53, Heb 2:2, see also Ps 68:17 (implied). We also see this in the case of John the Revelator in Rev 1:1, 2 where the so-called "chain of revelation" is: God to Jesus to Angel to John to people. Thus, an angel is involved in John's messages as well.
Last, Gal 3:19 tells us that this law given to Moses at Sinai, given by angels, was carried to the people by the hand of a mediator. This obviously occurred as Moses was given two tables of stone (Ex 25:16, 21, 31:18, 32:15, 18, 34:28, 29, 40:20, Deut 4:13, 9:9-11, 15, 1 Kings 8:9, 21, 2 Chron 5:10, Heb 9:4) which Moses then carried down the mountain and delivered to the people. [Note that the so-called "10 commandments" recorded in Ex 20:1-17, called "the covenant" (Ex 34:28, Deut 4:13) was really only a summary of the larger law as recorded in Ex 20:22-23:33 and later expand further in the book of Deuteronomy.]
Therefore, it appears that Paul is quoting well-known Hebrew history when he says that the law was "administered/ordained through angels by a mediator [ie, Moses]."
Brice L Martin, in Christ and the Law in Paul 2014, writes under 2.3 Origin of the Law:
In Romans and 1 Corinthians the law clearly originates with God. The law is given by God (Rom 9:4) and written by God (gegraptai, 1 Cor 9:9; 14:21; cf 14:34) and is called "the law of God" (7:22, 25; 8:7). Some have argued, though, that in Galatians the law does not originate with God but with the angels.109 The key text is Gal 3:19. In the process of contrasting the later law which cannot nullify the earlier promise Paul says the law ''was ordained by angels through an intermediary. Now an intermediary implies more than one; but God is one" (diatageis di’ angeldn en cheiri mesitou. ho de mesites henos ouk estin, ho de theos heis estin). Even here the view that the Torah did not spring from God is difficult to maintain. The preposition is dia not hypo; dia means not "by" but "through the agency of." The Torah stems ultimately not from the angels but from God. Diatageis, prosetethe, and edothe (vv 19, 21) should be viewed as Divine passives; God gave the law, he added the law after the promise, he ordained it through the agency (di’) of angels in the hands of a mediator (Moses). Also at Deut 33:2 (LXX), Josephus (Antiq. XV: 5: 3), Jubilees 1:27-29, Acts 7:53 and in Talmudic passages the presence of angels at the giving of the law is a mark of its excellence.
The preposition in Gal 3:19, showing most translations that use "by angels" instead of through, are technically wrong. The phrase is διαταγεὶς δια ἀγγέλων ἐν χειρὶ μεσίτου ordained through angels in the hand of a mediator. The word for ordained simply means to appoint, arrange, direct, command.
The angels act as agents and representatives of God; sometimes the distinction between God and his angel is blurred in that he acts as an angel. The ordination of the law was well established to have been done through the angels. James Dunn, writes in The New Perspective on Paul 2007, p.264:
As for the angels of Gal 3.19b, it is certainly the case, as already noted, that the phrase strengthens the contrast with the promise, the double phrase, ‘through angels’ and ‘by the hand of an intermediary’, heightening the contrast with the immediacy of God’s promise given to Abraham. But that said, it has also to be said that the reference itself does not imply that the angels in question were hostile or evil. On the contrary, anyone familiar with Jewish tradition would think most naturally of the well established Jewish belief that angels were indeed associated in the giving of the law (Deut. 33.2 LXX – ‘angels from his right hand were with him [the Lord]’; Jub. 1.29–2.1; Philo, Som. 1.143; Josephus, Ant. 15.136; Apoc. Mos. preface).20 And since the motif was also familiar elsewhere in Greek speaking Christianity (Acts 7.38, 53; Heb. 2.2), the most natural inference is that Paul intended the same allusion. In other words, the allusion, while marking a contrast with the promise, certainly did not deny the law was given from God – even if at one or two removes.
