Third marriage of king Władysław II Jagiełło (Jogaila) was his only marriage from love, and shortest one too. Jogaila's decision to marry Elżbieta, widow of castellan Wincenty Granowski, who died in 1410, likely poisoned by Teutonic Knights, was heavy criticized. She was not of royal blood (although she was closely related to most powerful Polish families, her mother Jadwiga of Melsztyn was godmother of Jogaila, her uncle Spytek II of Melsztyn was voivode of Cracow) and her age makes it unlikely that she could give heirs to the king.
Traditionally it was assumed, that Elżbieta was born in 1372, so she'd be 45 years old at the time of her first marriage, although now that date is questioned, just like unbelievable story of her first two marriages, which were likely fictional, invented by Jan Długosz, who disliked Jagiellons. Likely Granowski was her only husband before Jogaila and at the time of her marriage with the King she was 35-37 years old, not 45 (still, by medieval standards that is quite late for marriage for a woman, but there are still chances for offspring).

So lets stick with younger Granowska theory and let's assume she is healthier (she died after 3 years of marriage, so her health likely started to deterioriate around the time she wed Jogaila) so she lives extra 15 years and gives birth to two royal sons (Władysław b. 1418, Kazimierz, b. 1420).

So what would be consequences? Queen Elizabeth was unpopular, in large part due to concerns, that she would not be able to give Jogaila a heir, ITTL these concerns are proved to be false. Jogaila would, obviously, try to secure throne for his sons. That should be a bit easier than IOTL with sons of Zofia Holszańska-Zbigniew Oleśnicki, who IOTL fought fiercely to deny Jogaila's sons hereditary rights to Polish throne, is not yet bishop of Cracow (bishop of Cracow was at the time politically the most influental member of Church hierarchy in Poland, more influental than archbishop of Gniezno, due to fact, that Cracow was capital) so is not important enough now to challenge the King. Jogaila's sons are also few years older than IOTL and related to powerful Melsztyński and Tarnowski families (which has good and bad sides). So how would Jogaila's struggle for his sons' succession end?
@krieger
 
Władysław Jr would be 16 years old at the time of father's death, so no need for regency, but still be quite young at the time of Vytautas' death, unless Vytautas lives a bit longer (which seems possible if his coronation affair is butterflied away-humilitation which Vytautas suffered after his failed coronation plans could hasten his death, ITTL Jogaila likely has no reason to support Vytautas coronation attempt, so Vytautas would abandon them earlier, before they get too far and there was no way back), Casimir IV was made Grand Duke of Lithuania at age 13, so he was only one year older than Władysław Jr would be in 1430, so Jogaila could try to make his son Grand Duke after death of Vytautas.
 

krieger

Banned
Third marriage of king Władysław II Jagiełło (Jogaila) was his only marriage from love, and shortest one too. Jogaila's decision to marry Elżbieta, widow of castellan Wincenty Granowski, who died in 1410, likely poisoned by Teutonic Knights, was heavy criticized. She was not of royal blood (although she was closely related to most powerful Polish families, her mother Jadwiga of Melsztyn was godmother of Jogaila, her uncle Spytek II of Melsztyn was voivode of Cracow) and her age makes it unlikely that she could give heirs to the king.
Traditionally it was assumed, that Elżbieta was born in 1372, so she'd be 45 years old at the time of her first marriage, although now that date is questioned, just like unbelievable story of her first two marriages, which were likely fictional, invented by Jan Długosz, who disliked Jagiellons. Likely Granowski was her only husband before Jogaila and at the time of her marriage with the King she was 35-37 years old, not 45 (still, by medieval standards that is quite late for marriage for a woman, but there are still chances for offspring).

So lets stick with younger Granowska theory and let's assume she is healthier (she died after 3 years of marriage, so her health likely started to deterioriate around the time she wed Jogaila) so she lives extra 15 years and gives birth to two royal sons (Władysław b. 1418, Kazimierz, b. 1420).

So what would be consequences? Queen Elizabeth was unpopular, in large part due to concerns, that she would not be able to give Jogaila a heir, ITTL these concerns are proved to be false. Jogaila would, obviously, try to secure throne for his sons. That should be a bit easier than IOTL with sons of Zofia Holszańska-Zbigniew Oleśnicki, who IOTL fought fiercely to deny Jogaila's sons hereditary rights to Polish throne, is not yet bishop of Cracow (bishop of Cracow was at the time politically the most influental member of Church hierarchy in Poland, more influental than archbishop of Gniezno, due to fact, that Cracow was capital) so is not important enough now to challenge the King. Jogaila's sons are also few years older than IOTL and related to powerful Melsztyński and Tarnowski families (which has good and bad sides). So how would Jogaila's struggle for his sons' succession end?
@krieger

Elizabeth was also unpopular among Polish nobles because she was one of them before marriage. They despised her as a result of envy. It wouldn't vanish with a birth of heirs. And their rights could still be undermined. Elizabeth didn't have any royal blood (Polish or foreign) in her veins, so a lot of noblemen would resent her kids for being too "lowborn" to sit on the throne. Blood connections to Melsztyński/Tarnowski family (these houses were both descended from Spycimir Leliwita) could also be disadvantage under certain circumstances. We should also notice the fact, that Jogaila would be involved in struggle with nobility when he was younger than during OTL struggle. It means that he would be less willing to make a compromise with nobility and sign charter of priviledeges. I think that Jogaiła (supported by Melsztyński family and their allies) would not agree to any new priviledeges for nobility in exchange for regcognition of hereditary rights of his sons. It could lead even to a civil war.
 
Resentment to Granowska's relatives among nobility non related to her is obvious-that is a bit like situation in England with Elizabeth Woodville relatives (that is why it is not good idea to marry your subject if you're a monarch).
Still, Jogaila could play Lithuanian card and he would use it to bigger degree than IOTL (because he hardly have better arguments) and either his sons would be accepted or union would be broken.
 

krieger

Banned
Yes, Jogaila would play Lithuanian card. He could use Lithuanian troops to deal with opposition forcefully. And without Oleśnicki's diplomatic skills noble rebellion against Jogaila proclaiming his sons heirs would happen.
 
Civil war definitely gives TO some time, but their collapse was inevitable at that point.
Breakup of Polish-Lithuanian union would give Teutonic Knights a lot of time, although if King wins civil war their situation could be even worse than IOTL.
 

krieger

Banned
Breakup of Polish-Lithuanian union would give Teutonic Knights a lot of time, although if King wins civil war their situation could be even worse than IOTL.

I'm not sure about breakup. It would happen only if nobles defeated King in civil war. Otherwise, union would still continue.
 
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