"When the Aesir had refused to give satisfaction for the murder of Gullveig, and when Odin, by hurling his spear, had indicated that the treaty of peace between him and the Vanir was broken, the latter leave the assembly hall and Asgard. This is evident from the fact that they afterwards return to Asgard and attack the citadel of the Aesir clan (Völuspá; Saxo, Book 6). The gods are now divided into two hostile camps: on the one side Odin and his allies, among whom are Heimdall and Skadi; on the other Njörd, Frigg (Saxo, Hist., Book 1), Frey, Ull (Saxo, Hist., Book 3), and Freyja and her husband Svipdag, besides all that clan of divinities who were not adopted in Asgard, but belong to the race of Vanir and dwell in Vanaheim.
 
"So far as Skadi is concerned, the breach between the gods seems to have furnished her an opportunity of getting a divorce from Njörd, with whom she did not live on good terms. According to statements found in the myths, Thjazi's daughter and he were altogether too different in disposition to dwell in peace together. Saxo (Hist., Book 1.) and the Prose Edda (Gylfaginning 23) both have preserved the record of a song which describes their different tastes as to home and surroundings. Skadi loved Thrymheim, the rocky home of her father Thjazi, on whose snow-clad plains she was fond of running on skis and of felling wild beasts with her arrows; but when Njörd had remained nine nights among the mountains, he was weary of the rocks and of the howling of wolves, and longed for the song of swans on the sea-strand. But when Skadi accompanied him there, she could not long endure to be awakened every morning by the shrieking of sea-fowls. In Grímnismál 11, it is said that Skadi "now" occupies her father's "ancient home" in Thrymheim, but Njörd is not there named. In a strophe by Thord Sjareksson, we read that Skadi never became devoted to the Vana-god (nam-a snotr una goðbrúðr Vani), and Eyvind Skaldaspillir relates in Háleygjatal that there was a time when Odin dwelt í Manheimum together with Skadi, and begat with her many sons. With Manheimar is meant that part of the world which is inhabited by man; that is to say, Midgard and the lower world, where are also found a race of mennskir menn[1], and the topographical counterpart of the word is Asgard. Thus it must have been after his banishment from Asgard, while he was separated from Frigg and found refuge somewhere in Manheimar, that Odin had Skadi for his wife. Her epithet in Grímnismál 11, skír brúðr goða,[2] also seems to indicate that she had conjugal relations with more than one of the gods.

—1886, Viktor Rydberg Teutonic Mythology, No. 36

[1] mennskir menn, human men, living men; Lif and Lifthrasir in Mimir's grove.
[2] skír brúðr goða, the bright bride of gods. The word brúðr, bride, is a simple heiti for woman, therefore brúðr goða, divine bride, is a simple paraphrase for goddess. It does not necessarily indicate that Skadi has been the "bride" of more than one god.

Skáldskaparmál 3: “Now Skadi, the daughter of the giant Thjazi, took helm and birnie and all weapons of war and proceeded to Ásgard, to avenge her father. The Æsir, however, offered her reconciliation and atonement: the first article was that she should choose for herself a husband from among the Æsir and choose by the feet only, seeing no more of him. Then she saw the feet of one man, passing fair, and said: "I choose this one: in Baldr little can be loathly." But that was Njördr of Nóatún. She had this article also in her bond of reconciliation: that the Æsir must do a thing she thought they would not be able to accomplish: to make her laugh. Then Loki did this: he tied a cord to the beard of a nanny goat, the other end being about his own balls, and each gave way in turn, and the two screeched loudly; then Loki fell into Skadi's lap, and she laughed. Thereupon reconciliation was made with her on the Æsir’s behalf.” 
 
Skadi presents a unique situation. In surviving stories, Thor kills numerous giants including Thjazi, Geirrod, Thrym, Hymir, and Hrungnir. Thjazi, Geirrod and Hymnir are all said to have daughters, and Thyrm a sister. The tale of Skadi would seem to suggest that the daughter of any giant killed by the gods could demand compensation for the death of her father. Yet, remarkably Skadi is the only one who ever makes such a request. Not only does she demand it, but the gods willingly grant her request.  Why don’t the gods simply kill her as well? Thor has killed a great number of giantesses.
 
The answer, I believe lies in the identity of her father, Thjazi. Following the tale of Idunn’s kidnapping Snorri provides a brief geneaology of Thjazi, allowing us to learn more about him:
 
Skáldskaparmál 4: “It is so said, that Odin did this by way of atonement to Skadi: he took Thjazi's eyes and cast them up into the heavens, and made of them two stars.
“Then said Ægir: "It seems to me that Thjazi was a mighty man: now of what family was he?" Bragi answered: "His father was called Ölvaldi, and if I tell thee of him, thou wilt think these things wonders. He was very rich in gold.”
 
Snorri thus informs us that Thazi’s father was called Ölvaldi, who was “very rich in gold” and that his other sons were named Idi and Gangr. This information provides a thread for us to follow to learn more about this family of giants.
 
Thjazi is mentioned in several places in the lore. He appears by name in the Eddic poems Lokasenna, Harbardsljod, and the Grotti-song. In both the skaldic poem Haustlong and its prose retelling in Skaldskaparmal 3, we learn that Thjazi, the father of Skadi, once kidnapped Idunn from Asgard and concealed her from the gods. Without access to her golden apples, they grew old. It is for this that he was killed by the gods.
 
In Harbardsljod, 19, Thjazi’s father is called Audvaldi. There Thor throws Thjazi’s eyes into heaven.
 
Grotti-song 9 names the brothers Thjazi, Idi and Aurnir, identifying them as “mountain-giants.” Hrungnir is their kinsman.  
 

Since Idunn is Ivaldi’s daughter, we might suspect that her apples are a product of their forge. In the Rigveda, a group of smiths known as the Ribhus (a name etymologically associated with the Old Norse word alfr) also create treasures for the gods.   One of them is a means of rejuvenation, by which they restore their aged parents to youth. Without Idunn’s apples, the gods begin to grow old. Idunn is away for many years.

In time, they gather and recall that Loki was the last one seen with Idunn. They capture him and force him to find her. He flies to Jötunheim in Freyja’s falcon-dress. He enters Thjazi’s home while he is out fishing, find Idunn, and transforms her into a  nut. He grasps the nut in his claws and flies off with her. The giant sees them (perhaps because Loki was taunting) and follows in hot pursuit. Loki reaches Asgard first, and as Thjazi flies over the wall, the gods raise a “quick-fire” and singe his wings. Thor delivers the fatal blow with his lightning hammer and Thjazi is dead. Idunn is returned safely to Asgard and the gods rejoice. Until Skadi shows up in her war gear.

We are only told that Skadi is Thjazi’s daughter. The identity of her mother is unknown. Since Thjazi has been living with Idunn in Jotunheim, long enough for the gods to grow old without her apples, it’s entirely possible that Idunn is her mother. This circumstance alone would be reason enough for the gods to spare her and offer her a husband of her choosing from their number.  

In Hrafnagaldur Odins 6, we are told that Idunn, and thus her father and brothers are the kin of elves. The phrase “Aesir and Alfar” (elves) is a common formula found in several poems. We find an instance in an Old English charm.

In the lore, elves are usually depicted as archers. In Old English, disease is brought by “elf-shot”. In the poem Volundarkvida, the famous smith Volund, who is called an “elf-prince” uses skis and shoots bear. His brother is the famous archer Egil. Likewise, Skadi is called “the goddess of the bow” and “the goddess of skis” suggesting the same. Ull, the son of Sif and an unnamed father also shows similar characteristics, suggesting that Sif herself is of the same race.

After Thjazi’s death, the gods make stars of his eyes, as an act of atonement. This is the only time in the lore where Thor honors a giant he has killed. The only parallel to this story of star-making is that of Thor and Aurvandil, told by Snorri in Skaldskaparmal 25. There, Thor makes a star of his good friend’s toe, which was frozen off when he carried Auvandil on his back in a basket. Might Thor and Thjazi once have been friends? Perhaps before the gods had judged their works inferior to those of Brokk and Sindri? Before Loki had caused all this trouble and alienated both groups of artists from the gods.  

In Grímnismál 4 and 5, Thor is said to live close to Alfheim, and Frey is said to own it. The sons of Ivaldi make treasures for Thor, Frey and Odin. In Hymiskvida 7,8 (cp. 37, 38) we learn that on the way to Jotunheim, Thor leaves his goats at Egil’s house. In Volundarkvida, Egil is Volund’s brother, and therefore also an “elf-prince”.
Following the tale of Idunn’s abduction, Snorri tells a curious story about the Sons of Ölvaldi. He says that after their father’s death, Thjazi and his brothers divided his gold among them, taking equal shares of it in their mouth. Thus gold may be referred to as the speech of these giants. As proof of this, we find two kennings for gold in the partially preserved ancient poem Bjarkamál in fornu, quoted in Skáldskaparmál 45. They are Thjaza thingskil, “Thjazi’s testimony”, and Iðja glysmál, “Idi’s shining speech”.
 
Examining these kennings, we have good reason to doubt Snorri’s explanation. It seems as if it were created to explain the kennings for gold, Thjaza thingskil and Iðja glysmál. While the word mál in Idi’s glysmál can refer to ordinary speech, the word thingskil refers to testimony before a court (i.e. a “Thing”). Anthony Faulkes defines it as “assembly business”, “assembly declarations” (“Skáldskaparmál: Glossary and Index of Names”). Why would gold be designated as “Thjazi’s testimony before a Thing”? Do we know of any mythic circumstance in which golden works of art, made by Thjazi, serve as testimony in a court case?
 
If Ölvaldi-Allvaldi’s son Thjazi, is also one of Ivaldi’s sons, then the answer is yes. In the contest of the artists described above, the Sons of Ivaldi were not present, when their works were judged before a tribunal of the gods. Their golden works of art were mute testimony of their skill as artisans. Because Ivaldi’s sons were not present to speak for themselves, their work figuratively spoke for them. If Thjazi and Idi, Ölvaldi-Allvaldi’s sons, were considered identical to Ivaldi’s sons, then the kennings Thjaza thingskil, Thjazi’s testimony, as well as Iðja glysmál, “Idi’s shining speech” find their natural explanation. That gold which was held in the mouth should be referred to as thingskil is improbable.
 
That the heathen skalds regarded Ivaldi’s sons as identical to Ölvaldi-Allvaldi’s sons, should now be apparent. But can any of this be corroborated by any other source?
 
In Lokasenna 17, Loki insults Idunn by saying:
 

If Thjazi is indeed a smith, one of the famous sons of Ivaldi, then the name of his hall Thrymheim, may refer to the noise of a smithy. The word þrym- refers to the din of battle, specifically to the clanging of metal swords.