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:
- to put into a proper order or relationship, make arrangements, eg, Acts 20:13
- to give detailed instructions as to what must be done, order, eg, Matt 11:1; 1 Cor 7:17, 9:14, 11:34, 16:1; Luke 3:13, 8:55; Acts 7:44, 18:2, 23:31, 24:23, Gal 3:19, Tit 1:5.
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]."