Patterns in Chords

Currently working through Chapter 2, and the 2nd exercise deals with forming chords a perfect 4th/5th apart, as well as those a 2nd and 3rd apart.

Spending some time on the keyboard seems to indicate a pattern of sorts.

Chords a 4th above/5th Below
For the chords a 4th apart, in their root position (i.e. no inversions), all one has to do is move the 2nd and 3rd notes one tone up to get to the chord a 4th away, to what I think is the first inversion.

E.g.

C to F (CEG -> CFA, 2nd inversion of FAC)
Dm to G (DFA -> DGB, 2nd inversion of GBD)
Em to Am
F to Bdim 
G to C (GBD -> GCE, which is 2nd inversion of CEG)

Am to Dm


On the keyboard in the C Major Scale, with the left hand, it means moving the thumb and middle fingers one key to the right.

Chords a 5th above/4th below
Similarly, a pattern can be found for chords a 5th below.

E.g.
C to G (CEG to BDG -> 1st inversion of GBD)
Dm to Am (DFA to CEA -> 1st inversion of ACE)
so on and so forth....

Again on the keyboard, this is done by shifting the last and middle fingers on the left hand one key to the left.

Chords a 2nd above/below
Erm, no patterns here, since they are basically moving all the keys either one key to the left or right..

Chords a 3rd above
And here another pattern emerges
C to E (CEG -> BEG, which is the 2nd inversion if EGB)
Dmin to F (DFA to CFA, which is the 2nd inversion if FAC)
so on and so forth....

On the lefthand keyboard in CMajor, this is moving the last finger on key to the left.

Chords a 3rd below
The final pattern I noticed.
C to Amin (CEG to CEA, which is the 1st inversion of ACE)
Dm to Bmin (DFA to DFB, which is the 1st inversion of BDF)
so on and so forth....

So on the left hand keyboard in C Major, this is moving the thumb one key to the right.

Not sure how useful this is, but perhaps it could be down the road.

NOTE: This are my personal observations, and it could be correct, or incorrect. Use the information here at thy own peril.