From an earlier answer of mine

The main impetus for Napoleon's exile to Elba lay with the Russian tsar, Emperor Alexander I, not the Bourbons or any other member of the Allied coalition. After the Russian occupation of Paris and the French Senate's dethronement of Napoleon, Alexander I desired to appear magnanimous to his defeated foe. In a conference with Caulaincourt on 5 April, Alexander I told the French envoy that a Bonapartist regency was unthinkable and that the Emperor needed to abdicate unconditionally. But the Russian emperor reassured Caulaincourt that "[Napoleon] will be provided for; we will give him an independent state." After Caulaincourt suggested Corsica, Sardinia, and Corfu, the pair decided on the island of Elba which was now under Austrian territory.

Russia's two main allies, Austria and Britain, were quite taken aback by Alexander I's unilateral decision. Some British quarters favored a more distant exile to India (favored by Wellington) or in England as an honored prisoner. In a letter to his mistress Wilhelmine von Sagan , Metternich wrote:

If I had been here three days sooner, [the offer of Elba] wouldn't have happened. I tell you your fine emperor has made a lot of mistakes and behaved like a schoolboy escaped from his teacher. Now the teacher has returned, things will get better.

Castlereagh initially refused to countenance the Elba exile, but eventually relented for the sake of inter-Allied unity. Napoleon himself was not thrilled by Elba and actually proposed through intermediaries to Castlereagh that the British instead offer him an exile in England. But such schemes came to naught and Elba remained a fait accompli of the Russian Emperor.

This was part of the reason why Napoleon's escape from the island looks more obvious in hindsight. The Bourbons were not willing to pay for the subsidies Alexander I thought necessary for the deposed Napoleon. Nor was Napoleon willing to play the role of the Tsar's potential lieutenant in the future or play the role of a grateful supplicant. The island's proximity to the continent invited Napoleon's return and the Bourbons' general hostility to the Elba exile guaranteed it.

Although there were some calls for Napoleon's execution after the Hundred Days, especially within Prussian military circles, most of Napoleon's enemies recognized that an execution would not only make Napoleon a martyr, but would further complicate the transition from Napoleonic rule to what would later be known as the Congress of Vienna system.

Wellington's thoughts on this matter of execution provides a window into this mentality of the summer of 1815. In a letter written to Charles Stuart shortly after Napoleon's abdication, he wrote:

Blücher wishes to kill him [Napoleon]; but I have told him that I shall remonstrate, and shall insist upon his being disposed of by common accord. I have likewise said that, as a private friend, I advised him to have nothing to do with so foul a transaction; that he and I had acted too distinguished parts in these transactions to become executioners; and that I was determined that if the Sovereigns wished to put him to death they should appoint an executioner, which should not be me.

Wellington's letter is worth unpacking. Firstly, Wellington contended that Napoleon's fate was something that needed to be decided by "common accord," that is, by the concert of the various sovereign leaders of Europe then meeting in Vienna. Wellington's disgust at the type of revanchist sentiments as displayed by Blücher also signified that such feelings of revenge was relatively atypical. The Prussian army of the late Napoleonic wars was an independent-minded organization and sometimes pursued policies without authorization of its government or its king, such as the Convention of Tauroggen. Finally, Wellington also noted that while he would abide a decision by the various Concert leaders to execute Napoleon, he himself would not do so. Such a disapproval by this prominent general and political figure demonstrates how an execution would become a politically unpopular decision, even among the enemies of Napoleon.

One particular episode of the post-Waterloo history illustrates how politically charged executions of prominent Bonapartists could be: the execution of Marshal Ney. Under pressure from the Duchesse d'Angouleme and various emigres to clean house after the Hundred Days, Louis XVIII issued an ordonnance du roi of 57 figures to be tried under a courts-martial for their activities during the Hundred Days, including Ney. The government's first choice for the presiding judge, Marshal de Mouncey, refused on the grounds that:

Shall 25 years of my glorious labors be sullied in a single day? Shall my locks, bleached under the helmet, be only proofs of my shame? No, Sire! It shall not be said that the elder of the marshals of France contributed to the misfortunes of his country. My life, my fortune, all that I possess or enjoy is at the service of my king and country; but my honor is exclusively my own, and no human power can ravish it from me. If my name is to be the only heritage left to my children, at least let it not be disgraced.

The strong sentiment in the Restoration military for clemency led to a trial in the Chamber of Peers, which was dominated by emigres. During this trial, current Minister of War Marshal Gouvion St. Cyr appealed for clemency and so to did several of Ney's colleagues who had found themselves a new role in the Bourbon military. Giving into the revanchist sentiments of d'Angouleme threatened to destroy the fragile political order of the Restoration, which tried to knit together the Napoleonic and Ancien Regime's two aristocracies. Ney's execution also met with wider disapproval in other European circles. In Britain, Lord Holland championed clemency for Ney because it was a violation of Convention of Paris, which was an armistice that promised no political retributions in exchange for peace and the surrender of Paris. Politics were at play here too, as Lord Holland used Ney's execution to attack the rising political star of Wellington by asserting that the British general was seeking revenge upon an enemy he could not defeat in the field. The resurgent British radical press also saw the Bourbon executions as further evidence that the existing political system was rotten and needed radical reform.

Although it is conjecture, if Ney's execution could prompt such sentiments, it is a reasonable proposition that an execution of Napoleon would create an even bigger political tempest. Napoleon was such an important and outsized symbol for an era that execution would be a dangerous undertaking, but Napoleon needed to be kept under close scrutiny and be distant from European politics. The various Allied leaders had actually recognized this problem prior to the Hundred Days and there were tentative steps towards exploring the possibility of exiling Napoleon in some remote location. Both Castlereagh and Metternich felt that Elba was too proximate to Europe to be a gilded cage, but accepted Elba as a means to placate Tsar Alexander I who wished to appear magnanimous to a defeated foe. The Hundred Days gave advocates of a distant exile a iron-clad pretext to shuttle Napoleon off to some distant shores. Wellington favored Madras in India, but St. Helena was an ideal choice. The fear of Napoleon's personal charm and image was still powerful even in defeat. When Napoleon was likening himself to Themistocles on HMS Bellerophon, Castlereagh was scared that Napoleon would have had an undue influence upon the Prince Regent, who maintained a Bonapartist style at court and it was feared could use ill-treatment of Napoleon for his own political capital by likening such cruel imprisonment to the treatment of his father, George III.

As the above indicates, Restoration political culture was was both highly contentious and Napoleon was not just a single man in it, but a symbol for his age. Executing Napoleon would have entailed a wider destabilization of European politics when it was in the interest of the Congress's leadership to try and stabilize the political landscape. A large number of Napoleonic officials retained powerful positions in Restoration society, like Gouvian St. Cyr or Eugène de Beauharnais, and many of those Frenchmen immediately exiled after the Hundred Days like Soult shortly returned and became important parts of the establishment. At a lower level, many of the Napoleonic elite were cementing themselves as the new elite of a post-Napoleonic France. Further afield in the realm of culture, Napoleon quickly emerged as a pan-European symbol for Romanticism and the defeat of such a titan was an important plank for legitimizing the various post-1815 regimes. Within this context, a swift drumhead execution, which as the wrangling over Ney's trial demonstrates would have likely been the only viable way to execute the emperor, would have tarnished the victory over such a giant and even the Bourbons recognized this.

Sources

Bew, John. Castlereagh: A Life. New York: Oxford University Press, 2012.

Dwyer, Philip G. Citizen Emperor: Napoleon in Power.New Haven : Yale University Press, 2013.

Millar, Stephan. " 'Pour encourager les autres': The Trial and Execution of Marshal Michel Ney." Napoleon-Series, http://www.napoleon-series.org/research/biographies/marshals/c_neyexecution.html .

Muir, Rory. Wellington. Volume 2: Waterloo and the fortunes of peace : 1814 - 1852. New Haven: Yale Univ. Press, 2015.

Price, Munro. Napoleon: The End of Glory. New York: Oxford University Press, 2014.