It's an impulsion, not a case of showing off, and that's true for both of us.
From the guardian.co.uk
Impulsion can occur at the walk, trot and canter.
From the en.wikipedia.org
Dynamic imagery scored the highest for all measured aspects of the jump including approach, curve, impulsion, and bar clearance.
From the sciencedaily.com
There can be majesis by impulsion.
From the guardian.co.uk
The horse has good, solid impulsion.
From the en.wikipedia.org
The scores for the general attributes of gait, submission, impulsion, and rider performance mentioned above are scored using a coefficient.
From the en.wikipedia.org
For example, a crooked horse cannot develop impulsion, and a horse that is not relaxed will be less likely to travel with a rhythmic gait.
From the en.wikipedia.org
Singh's migration to the city, a combination of divine impulsion and the simple need to work, is part of what could be called an epic trend affecting billions of people worldwide.
From the abcnews.go.com
In addition to marks for the dressage movements, marks are also awarded for more general attributes such as the horse's gaits, submission, impulsion and the rider's performance.
From the en.wikipedia.org
More examples
Drift: a force that moves something along
Impulse: the act of applying force suddenly; "the impulse knocked him over"
Impulsion is the pushing power (thrust) of a horse, which comes from his desire to move powerfully forward with energy. However, speed does not create impulsion, and a rushing horse is more likely to be "flat" than impulsive. ...
The act of impelling or driving onward, or the state of being impelled; the sudden or momentary agency of a body in motion on another body; also, the impelling force, or impulse; Influence acting unexpectedly or temporarily on the mind; sudden motive or influence; impulse
Thrust. Releasing of the energy stored by engagement. In dressage, impulsion is associated with a phase of suspension such as exists in trot and canter, but does not exist in walk. Therefore, impulsion is not applicable to the walk. ...
The response of a horse that is correctly trained in its go/stop responses so that it moves forward energetically with a self-maintained rhythm, straightness and outline when signalled to do so. ...
This is the controlled energy that is generated through the hindquarters and allows for engagement of the hindquarters to occur, the horse will maintain rhythm, suppleness and balance.
Strong, but controlled, forward movement in the horse (not to be confused with speed).
The driving force of a horse's back legs and hindquarters.