Why did Stalin exile Trotsky instead of killing or imprisoning him?

Question

Leon Trotsky, one of the original founders of the Bolshevik Revolution, was strongly opposed to the leadership and policies of Joseph Stalin. In response to Trotskys public criticism of him, Stalin banished him from the Soviet Union.

In 1924, Lenin died, and Joseph Stalin emerged as leader of the USSR. Against Stalin’s stated policies, Trotsky called for a continuing world revolution that would inevitably result in the dismantling of the Soviet state. He also criticized the new regime for suppressing democracy in the Communist Party and for failing to develop adequate economic planning. In response, Stalin and his supporters launched a propaganda counterattack against Trotsky. In 1925, he was removed from his post in the war commissariat. One year later, he was expelled from the Politburo and in 1927 from the Communist Party. In January 1928, Trotsky began his internal exile in Alma-Ata and the next January was expelled from the Soviet Union outright.

(Source: History Channel)

This seems odd. Stalin had no qualms about arbitrarily executing opponents (he even executed Trotsky's brother in law) Why would he exile one of his harshest critics instead of either killing him - or at the very least keeping him alive but imprisoned? What other factors were at play here ?

Answer 1

According to Trotsky (excrept from Коминтерн и ГПУ, published in Бюллетень оппозиции (большевиков-ленинцев) № 85, translation mine):

In 1928 […] not only the shooting squad, but even arrest would be inconceivable: the generation, alongside which I went thorugh the October Revolution and the civil war, was still alive. Politburo felt it was under siege from all sides. From Central Asia I had the ability to keep in constant contact with the growing opposition. In these conditions Stalin, after a year of doubt, decided to use exile as a lesser evil. His arguments were: isolated from USSR, with no staff or material resources, Trotsky will be unable to act […]. Stalin admitted on multiple occasions that my exile was his "greatest mistake".

He simply was too popular at the moment. Even his deportation to Almaty in January 1928 was handled in a very roundabout way - sure, he was forcibly removed from Moscow, but on the other hand - he was allowed to keep most of his posessions (most importantly - his vast personal archive), and until October he had full freedom in communication. On another hand, keeping him in Soviet Union was not an option, either: he kept on rallying his supporters. So he was exiled.

P.S. This isn't really related to the decision-making process regarding the exile, but: what I found interesting while researching this question was that even when Stalin consolidated his power he was mainly concerned by Trotsky's archive (in fact, the Stalin's "greatest mistake" remark Trotsky mentions was about allowing him to take the archive with him). The order for Trotsky's murder only came in 1939, when Trotsky in his article "Сверхборджиа в Кремле" publicly denounced Stalin for the pact with Germany (and also pretty much implied Stalin poisoned Lenin).