It wasn't easy, but I've found two solid references for this work.
The first one attributes it to the President's Council on Physical
Fitness and Sports at the time when George Allen headed up that group.
A second reference, in slightly revised form, attributes the
authorship directly to George Allen.
Several links to the relevant works can be seen here:
in the three files that begin with [ George Allen ]:
[ George Allen quote cited.jpg ] is an excerpt from the book:
Helping At-Risk Youth Through Physical Fitness Programming
Thomas R. Collingwood
Human Kinetics Publishers
You can see that the book refers to a poster created by George Allen,
and give a reference number (35) for the work. The reference itself
is shown in another file on the esnips page -- [ george allen
reference 35.jpg ] -- and it gives the full reference as:
35. President's Council on Physical Fitness and Sports. 1983. "What
is a workout?" Washington, DC: Government Printing Office.
A second version of the inspirational text appears in slightly
modified form directly under George Allen's byline as author. It is
called "What is Exercise", and can be seen in the file [ George Allen
Heartcorps.jpg ] (click on the "Original Size" button at the top of
the page to see it clearly).
The reference here is given as:
What is exercise? George W. Allen. Heartcorps. Jan-Feb 1989, v1 n4 p26(1).
Heartcorps was a magazine publication that appears to no longer be in print.
I trust this information fully answers your question.
However, please don't rate this answer until you have everything you
need. If you would like any additional information, just post a
Request for Clarification to let me know how I can assist you further,
and I'm at your service.
All the best,
search strategy -- searched Google, a9.com, shop.ebrary.com, and
several subscription databases for excerpts from the quote, including:
[ "what is a workout" ]
[ "one part physical exertion" ]