The LOTGK says she "salutes" the giants which suggests they either outrank her in power or are equally powerful in their own realms.

"salute" at its simplest means greet. 

Lewis uses its medieval meaning, which can involve a kiss of greeting, as well as a mark of respect. It comes from Latin word, which has a suggestion of wishing the person good health. 

The LOTGK has a good relationship with giants, and perhaps she addresses them in a respectful way, but I doubt she has respect for them.

...Ok I know this theory has already been addressed and dismissed, but I do have a personal soft spot for the possibility that she's Jadis' daughter, taking revenge for her mother. As to who would be willing to have a child with an ice witch - Jadis is repeatedly said to be beautiful, and also I could see many evil-alligned creatures caring more about sucking up to power than having a *good* mate?

Another possibility I would be willing to accept, re: her and Jadis being of "the same crew", is the possibility that Jadis trained and indoctrinated a cult of witches of other species origins while she was in the North, and LOTGK is the last survivor or descendant of this cult.

Another thought I've run into that I find amusing, although less resonant than the previous possibilities, is that LOTGK is actually snakey rather than humanoid by her birth nature, and a witch took her in as a familiar/assistant and taught her to shapeshift.

@silverlily Interesting ideas! I don't think it's completely "against canon" that the Green Witch could be a family relation of Jadis — it's just that they couldn't be the same person, going by what we know from the books. But "one of the same crew" could very well mean they are literally related. Going by the very little we know, it's not outside the realms of possibility that Jadis could have had a daughter, perhaps fathered by one of the other evil creatures she may have met during her time in the North before she re-invaded Narnia as the White Witch. Maybe the Green Witch's father was also a humanoid who could turn into a snake, or a snake who could turn into a humanoid...

Or else (given the stretch of over 1,300 years between Jadis's final defeat in LWW and the Green Witch's reign in SC), maybe the Green Witch could be a later descendant — a granddaughter or further down — since we don't know whether she herself was immortal in the way Jadis became after eating the apple. But again, there's nothing to suggest they couldn't have been related "by blood", or that they didn't know each other in the distant past, if the Green Witch was also immortal (unless killed in battle) or at least very long-lived.

Posted by: @silverlily

Another thought I've run into that I find amusing, although less resonant than the previous possibilities, is that LOTGK is actually snakey rather than humanoid by her birth nature, and a witch took her in as a familiar/assistant and taught her to shapeshift.

I'd never thought of that idea, but I reckon it's not outside the bounds of possibility either! I sometimes wish Lewis himself was still around for us to bounce all these questions off him and I wonder what he'd say — hopefully something more interesting than "Oh, gosh, even I don't know — I never thought about it that far myself..."   (He was very happy to answer young fans' questions about Narnia when they wrote to him during his lifetime, and there's a book of his letters to children, but unfortunately I don't think any of them deal with this particular question!)

@spalatin-rengriss 

Came on this looking for anyone that noticed the connection between the Lady of the Green Kirtle and Morgan Le Fay.   It hit me when reading "Connecticut Yankee in King Arthur's Court" a book in Lewis's library.   What struct me initially was the contrast between her beauty and sweetness of personality and her evil nature.  But also they are both trilling and musical.   Here's the passages from Connecticut Yankee: 

To my surprise she was beautiful ; black thoughts had failed to make her expression repulsive, age had failed to wrinkle her satin skin or mar its bloomy freshness. She could have passed for old Urien's grand-daughter, she could have been mistaken for sister to her own son.
...
But Morgan was the main attraction, the conspicuous personality here; she was head chief of this household, that was plain. She caused us to be seated, and then she began, with all manner of pretty graces and graciousnesses, to ask me questions. Dear me, it was like a bird or a flute, or something, talking. I felt persuaded that this woman must have been misrepresented, lied about. She trilled along, and trilled along, and presently a handsome young page, clothed like the rainbow, and as easy and undulatory of movement as a wave, came with something on a golden salver, and kneeling to present it to her, overdid his graces and lost his balance, and so fell lightly against her knee. She slipped a dirk into him in as matter-of-course a way as another person would have harpooned a rat !

Poor child, he slumped to the floor, twisted his silken limbs in one great straining contortion of pain, and was dead. Out of the old king was wrung an involuntary " O-h !" of compassion. The look he got, made him cut it suddenly short and not put any more hyphens in it. Sir Uwaine, at a sign from his mother, went to the ante-room and called some servants, and meanwhile madame went rippling sweetly along with her talk.
...
Morgan le Fay rippled along as musically as ever. Marvellous woman.

Since Lewis was influenced by the King Arthur stories it is very likely that he would create characters that were based on those in the legends of Arthur and works of literature which contained similar stories such as Spenser’s Faerie Queene.  Morgan Le Fay would be an interesting character to base a witch on, and she was quite similar to the Lady of the Green Kirtle. The King Arthur stories have much in common with medieval Narnia. I think the character of the Lady worked very well in The Silver Chair. 🙂

Do people still read Mallory’s Le Morte d’ Arthur and Spenser’s Faerie Queen?  I think English majors in colleges and universities still read it, but it is no longer required reading for the basic courses in liberal arts institutions. So chances are that many people living here in the U. S. may not even know who Morgan Le Fay was.  It is kind of unfortunate that people’s knowledge of literature is rather poor, but it is the time we live in. They would know that the Lady of the Green Kirtle was a witch, but they may not make the connection with the medieval stories of King Arthur.  C. S. Lewis would have thought our culture is sadly lacking in its understanding of mythology. 🙁

I recently discovered another likely inspiration for LOTGK... this has apparently been mentioned on the forum in the (somewhat distant) past when the naming contest for the Lady was happening. 

I actually came across this painting:

https://www.google.com/search?q=lamia+painting&client=ms-android-samsung-rvo1&sca_esv=68b5c75d01228b39&ei=aum5ZvqKBanhwN4PwNqBWA&oq=lamia+paintinf&gs_lp=EhNtb2JpbGUtZ3dzLXdpei1zZXJwIg5sYW1pYSBwYWludGluZioCCAAyBxAAGIAEGA0yCBAAGAUYDRgeMgcQIRigARgKMgcQIRigARgKMgsQABiABBiGAxiKBTILEAAYgAQYhgMYigUyCxAAGIAEGIYDGIoFMgsQABiABBiGAxiKBUjUK1DSDliwJ3AAeACQAQCYAd8CoAH6C6oBCDExLjIuMC4xuAEByAEA-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&sclient=mobile-gws-wiz-serp#vhid=KhdwdlWwBC4K_M&vssid=_cem5Zq2qG-7jwN4Pgc_q8Ag_39

It shows a young knight being wooed by a lady in a garden... and if you look closely you'll see a recently-shed snakeskin on the ground, ominously hinting what this lady has been up to. The painting, by John Waterhouse, depicts the Greek tale of Lamia, a half-woman, half-snake entity said to ensnare heedless young men with her false beauty before turning into a serpent and preying upon them.The painting itself is inspired by John Keat's poem, also entitled Lamia, which Lewis was almost certainly familiar with (he apparently was such a fan of Keats he had been heard to shout at a student who disliked him: "If you think that way about Keats you needn't come here again!").

While I still think Error from The Faerie Queene had something to do with Lewis' inspiration here, this seems like a very direct influence. The imagery of the painting could practically be an illustration from the Silver Chair (although Lamia should admittedly be wearing green), and the poem is definitely in Lewis' wheelhouse.