“by loving.” The Hebrew can be translated “by loving,” which makes very good sense in English (cp. CEB; Geneva Bible (1599); NAB; NASB; NRSV; TNK; The Shocken Bible). Deuteronomy 30:20 tells us how to make the choice to live (see commentary on Deut. 30:16; 30:19).
“for he is your life.” The parenthetical reminder is that Yahweh is the true source of life. No one should think that if they keep a certain set of rules and regulations they will live successfully here on earth and live forever later. All life comes from God, and obeying God and walking in His ways is not to be a rote mechanical process, but the outward demonstration of a living and loving relationship with the Living God. In Jesus’ day, the Pharisees kept the commandments of the Law but did not understand the heart behind the Law and thus omitted things that represented the heart of the Law such as love, mercy, and justice (Matt. 23:23; Luke 11:42).
The Hebrew text can also be translated “for THAT is your life,” instead of “for he is your life,” which makes the verse say that loving Yahweh, obeying Him, and holding fast to Him is the source and purpose of life. Although a number of English versions and commentators support that reading (NASB; NLT; NRSV; RSV), and although it is certainly possible that the Hebrew text was written such that it could have—and indeed might have—both meanings, it seems that the primary meaning is that God is the source of life and blessing. Omitting the relationship with God from obeying God can lead to Pharisaical behavior; just doing the action without the relationship. The parenthetical statement keeps God in the picture, and the YLT is another version that has the parenthesis.