When studying an instrument, working with arpeggios is standard practice. This involves running triads, dominant and minor seventh chords, diminished and augmented and, eventually, extended chords. As instrumental practice, this is usually done mechanically, without absorbing the structure. In order to improvise, the mind and the fingers require constant communication with each other. The best approach is to not read out of practice books but to improvise different inversions.
Inversions take place in all chords but starting on different notes of the chord.
It is necessary to thoroughly understand this approach as you are doing it. There are ways of utilizing triads in extended harmony that involve slash chords which are shortcuts to playing extended intervals such as a triad over a bass tone or triad over a seventh chord. Whole systems of harmonic navigation can be developed this way. What the player is dealing with are sounds. A triad is a basic voicing.