The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post Suggestions for good notation. The difference is that I would be interested especially in the practical impact of the improved notation, i.e. examples that have actually created a better understanding of a given topic, or have advanced actual research work on a given topic, or communication about results.
I would be interested in two aspects in particular
(1) Visibility and efficiency: Improved and simplified notation that made structures and properties more clearly visible, and that created efficiencies (e.g., using less space and needed less time, dropped unnecessary or redundant details).
(2) Evolving process: Mathematical work is constantly creating new objects, i.e. in constant need of new symbols, or using symbols in new ways. And it needs to create a shared understanding about that. Would you have any practical examples of this evolving process, including dead-ends and breakthroughs?
I would be most interested to learn about practical examples, and would be grateful if you were willing to share.