Step 1: Understanding the Question:
The word in the question is "projected". It means the image or story that Indira Gandhi gave to the public. It is different from what really caused the split.
Step 2: Separate cause from projection.
The real cause was a power struggle between Indira Gandhi and the Syndicate. The Syndicate had made her Prime Minister in 1966 and expected to guide her. The presidential election of 1969 brought the quarrel into the open.
Step 3: Look at her public message.
In public she spoke about bank nationalisation and the abolition of privy purses. She said these steps were opposed by the old guard. So she turned a personal fight into a fight of ideas.
Step 4: Remove the wrong options.
Option (1) describes the immediate event, the Whip not being obeyed, and not her projection. Option (2) fails on the dates because the 1967 elections were already held. Option (3) says the split removed differences, but a split increases them.
Step 5: Pick the match.
Only option (4) talks about ideas. Socialist against conservative is the exact label used by her side at that time.
Final Answer:
The projection was an ideological divide, so option (4) is correct.
\[ \boxed{\text{Option 4}} \]