What clause in the roman republics political organization presented its greatest danger?

The most dangerous executive post during the Roman Republic as the appointment of a dictatorat times of emrgencies. This was an extraordinary officer of state who was not legally liable for his actions and had an absolute authority which went beyond that of the consuls (the heads of the Republic) who were ordinary officers of state and had legal liability for their actions. A dictator could overrule or depose and sentence to death other officers of state. He could rule by decree and change any law, but these lasted only for his term of office. He could also issue new laws without the assemblies' vote, but these were usually put through a vote. He could punish without trial and was exempt from appeal. He could act without consulting the senate, but he usually did. The Word did not have the negative connotation it has now and did not imply despotism.

The term of the dictator was only six months as the office was only for dealing with emergencies. His mandate was established by senatorial decree issued prior to his appointment and was defined by and restricted to the cause of the appointment. It was rei gerundae causa (for the matter to be done). The most common cause was a military emergency. The dictator was given sole lead of the army. This was done to make military command more efficient or to appoint a man with better military leadership skills. Occasionally, a dictator was appointed seditionis sedandae - for putting down rebellion.

This practice was stopped (with the exception of two instances in the second Punic war) after the wars in Italy because it was feared that a man with such great power would be dangerous if he was far from Rome. It was replaced by the Senatus consultum ultimum, a senatorial emergency decree that allowed the consuls to act as they saw fit. The Romans preferred to do this or to re-elect a consul despite a law that prohibited re-election for 10 years. Gaius Marius was elected consul six times, five of which were consecutive (107, 105-100 BC) because Rome had to deal military emergencies.

Sulla was appointed dictator (legibus faciendis et reipublicae constituendae causa, (for the making of laws and the settling of the constitution) at the end of his war against Marius (82 BC). He had thousands of his political opponents executed. It was this that gave the Dictatorship a bad name. In 53 BC, the senate, mindful of Sulla' precedent, appointed Pompey sole consul instead of dictator, to deal with domestic unrest. This made him accountable for his actions at the end of his office. Julius Caesar was appointed dictator four times.