Your Left Elbow forms part of the putting triangle
recommended for the putting stroke.
How you position your elbows throughout your putting stroke is a matter of preference. Your left elbow can be held into your left side or allowed to separate during your follow-through.
There is no right or wrong way to arrange your elbows at address other than to say that they should be free of tension.
While it is probably a good idea to keep your Right Elbow close to your side throughout your putting stroke, you can have some separation of your left, both at setup, and during your follow-through.
The legendary player Horton Smith, author of the book The Secret of Holing Putts, and renown in the 1930s and '40s for his putting ability, would deliberately aim his left elbow at right angles to the path of his putt as part of his pre-putt routine.
In putting the analogy is made that the stroke is a nothing more than a miniature swing. Stan Utley states in his book The Art of Putting that, in his opinion, "the putting stroke shouldn't be fundamentally different from a full shot, just a smaller version of it."
In this is the case then it would suggest that any form of disengagement of your elbows is poor technique. However, there are too many examples of good putters, past and present, whose stroke doesn't resemble a mini-swing, to make a water-tight case.
Pictures of Jack Nicklaus show that his left elbow is away from his body at address.
Todd Sones in Lights-Out Putting writes that the most accurate way for most amateurs to putt is to allow their arms to separate from their hips. In contrast, Greg Norman in his book Advanced Golf describes how his posture allows him to let his arms hang down naturally with his elbows close to his body throughout the putt.
So who is right and who is wrong? The answer is no-one as every golfer has to discover through practice which way of putting allows him or her to putt in the most consistent way.
For example, Dave Stockton allows his left to separate well away from his side in his forward stroke as the back of his left hand continues directly down his aimline (target line).
The extent of the separation, if any, is therefore a matter of preference. Attempting to consciously holding it to your side during your follow-through could lead to you rotating your shoulders and closing down your putterface prematurely through the contact area.
The secret of accurate putting is retaining the squareness of your putter blade to your aimline through impact and a bit beyond.
If you can do this consistently, well and good. If not, you can experiment to see if some modification can lead to an improvement.
1 = The Complete Short Game - Ernie Els
2 = The Secret of Holing Putts - Horton Smith
3 = Dave Stockton's Putt to Win