How does Gawain exhibit loving kindness in Sir Gawain and the Green Knight?
Gawain is one of the most tactful knights ever to grace King Arthur's Round Table. Gawain is hailed as the most gracious knight in all of England and shows that with everyone he meets. He follows the rules of knighthood to a "T" and never misses a step. The first time we are introduced to his "loving kindness," as you say, is when he steps up for Arthur against the Green Knight's challenge. But it's not just the fact that Gawain steps up for Arthur that shows his kindness, it's what he says that is so impressive.
"I beseech ye, my lord, let this venture be mine. . .For I think it not seemly when such challenges be made in your hall that ye yourself should undertake it, while there are many bold knights who sit beside ye, none are there, me thinks, of redier will under heaven, or more valiant in open field. I am the weakest and the feeblest of wit, and will be the less loss of my life if ye seek sooth"(5).
He presents himself as the weakest and easiest loss to the court, thereby permitting Arthur and the other knights to save face and not be embarrassed for not stepping up sooner before the Green Knight could laugh at them.