"but after the spirit, by which is meant, not spiritual worship, in opposition to carnal ordinances; but rather, either a principle of grace, in opposition to corrupt nature, called "Spirit", from the author, subject, and nature of it; or the Holy Spirit of God, the efficient cause of all grace: to walk after him, is to make him our guide, to follow his dictates, influences, and directions; as such do, who walk by faith on Christ, and in imitation of him, in the ways of righteousness and holiness; and such persons walk pleasantly, cheerfully, and safely: now let it be observed, that this walk and conversation of the saints, is not the cause of there being no condemnation to them; but is descriptive of the persons interested in such a privilege; and is evidential of their right unto it, as well as of their being in Christ: and it may be further observed, that there must be union to Christ, or a being in him, before there can be walking after the Spirit."
- from John Gill's Bible commentary on Romans 8:1