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Alexander Jagiellon was't very fortunate ruler, so let's improve his reign a bit.
ITTL Alexander is more capable, healthlier and more lucky. Let's start with his marriage: Helen of Moscow gives him a son in 1498 and dies in childbirth (Alexander would not consider it fortunate event, he really loved his wife, but objectively it would be better for him in the long run, Helen's Orthodox faith was obstacle for Alexander when he was candidate to Polish throne-Queen mother and Catholic hierarchy despised her).
-In 1501 Alexander does not behave like IOTL, when he left Lithuania in hurry taking Lithuania's treasure to Poland during war with Moscow and abandining his allies-Grand Master of Livonian branch of Teutonic Order and Khan of Golden Orde. Grand Master and Khan were meant to attack Moscow from north and south, while Lithuanians would take care of the center. IOTL Tatars and Teutonic Knights achieved nothing and had to retreat because Lithuanians, who were meant to be main force, abandoned them-their ruler was busy in Poland seeking for Polish throne. That was OTL. Say that ITTL Alexander assumes that Polish throne could wait and that victory against Moscow would improve his position, so he does not betray his allies and combined Livonian-Lithuanian-Tatar forces are victorious, Alexander is able to restore Lithianian border from beginning of his reign. Thus victorious Alexander is not desperate like IOTL and does not need to agree for Union of Mielnik and Priviledge of Mielnik. Lithuanian position towards Poland would be better than IOTL, position of monarch too.
-Alexander is healthier, thus he lives longer. That means Mikhail Glinski is not going to rebel and escape to Moscow. Glinski was Alexander's personal friend (unlike Sigismund, who did not knew Gliński). Without any doubt Alexander would take side of Gliński in his conflict with Zabrzeziński. Gliński would not be accused of murder and would not need to escape. Thus we have no Glinski family in Moscow and Ivan IV butterflied away.
-Sigismund either remains governor or Silesia or he may try his luck in Sweden (IOTL Swedish rebels proposed him Swedish crown twice, first time in 1506 or 1507). Alexander would like Sigismund in Sweden IMHO, otherwise Sigismund would be potential competitor to Alexander's own son.
With stronger Lithuania something like Union of Lublin is not going to happen, edpecially if Jagiellon line continues, and Poland is not that dominant partner of the Union, although despite Alexander's victory, Moscow still is stronger than Lithuania, thus Lithuanians could not afford to break their ties with Poland completly.
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