In these last few days, many of the companions explicitly disregarded the prophet's instructions in order to secure political power after his demise. The prophet had attempted to prevent their schemes by instructing many of the senior companions to leave Medina with Usama ibn Zayd, so that Imam Ali could assume leadership without interference, but many delayed and made excuses.
Some hypocrites were known, but others were skilled at hiding their hypocrisy as proven by the failed assassination attempt.
In the even known as "The Calamity of Thursday", while Prophet Muhammad lay on his death bed he decided to take an extra step to explicitly document the his successor and asked for something to write to leave a message "after which you'll never go astray."
However, a group of companions who were around him at the time, including Umar, realized that the prophet was planning to document his succession plan and started to strongly object and shout, prompting Prophet Muhammad to order them to leave.
The lecture analyzes the various Sunni sources which indicate that the opposition group, which Umar was part of, started saying prophet was "going insane."
The fact that these people were willing to sling mud on the prophet's sanity showed to the prophet the futility of writing down his will later. If he were to do so, those people would cast the same aspersions on the prophet's will, casting doubts on his sanity during every instance where he designated Imam Ali as his successor. This could be potentially stretched back all the way to revelation, and the spread of such slander would put people's belief in Islam itself at risk.
Lecture notes available at http://www.why-quran.org/?p=796.
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Video recording of this lecture + Q&A available on YouTube at https://www.youtube.com/watch?v=yBxvKOHjZcM&list=PLpkB0iwLgfTat-Pgh4W3WFmupPamiC9UT.