TITLE: New notation yielded new insights into identities
QUESTION [5 upvotes]: It is well known that using different notation can significantly reduce the proof of identities or other properties. Has there been cases of using a new notation led to new insights to yet unknown identities or properties?

REPLY [3 votes]: Three other examples :

The dot-notation for ... dot-product (or scalar product, or "interior product" by contrast with "outer product"), making notations using vectors like this one

$$(\vec{u}+\vec{v})^2=\vec{u}^2+\vec{v}^2+2 \vec{u} \cdot \vec{v}$$
analogous to their algebraic computation counterpart rules on numbers
$$(a+b)^2=a^2+b^2+2a . b.$$
This analogy helps to find new identities : for example, we are fully confident that
$$\vec{u}^2-\vec{v}^2=(\vec{u}+\vec{v}) \cdot (\vec{u}-\vec{v})$$
holds because its ordinary algebraic analog is true.


Quaternions' multiplication:

Instead of the detailed development
$$qq'=(a+a_1i+a_2j+a_3k)(a'+a'_1i+a'_2j+a'_3k)=aa'+...+a_3a'_3k^2$$
one can group the "real" and "vector" part under the form,
$$qq'=(a+V)(a'+V')$$
$$qq'=(aa'-\underbrace{V \cdot V'}_{\text{dot product}})+(aV'+a'V+\underbrace{\color{red}{V \times V'}}_{\text{cross product}})$$
underlining a close analogy with multiplication of two complex numbers (set apart the red term accounting for the non-commutativity of quaternion product). See formulas (4) and (5) in this answer.
A big thank to Mr. Gibbs, the physicist who introduced the dot product and the cross product notations around 1900.


The blocks notation for matrices.

Let us for example a $2 \times 2$ block decomposition :
$$M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$$
where the only condition is that blocks $A,B,C,D$ must have compatible dimensions.
This notation helps to simplify the description of many matrix operations, and provides often deep hints.
For example, if $A$,$B$,$C$,$D$ are all square blocks:
$$\det(M)=\det(A)\det(D)-\det(B)\det(C)$$
is true under a certain condition ($C$ and $D$ have to commute).
Another example, the "augmentation" of a quadratic form :
If $A$ is an $n \times n$ matrix, if $U,X \in \mathbb{R^n}$ and $b,y \in \mathbb{R}$, we can expand the following quadratic form  :
$$(X,y)^T\begin{pmatrix}A&U\\ U^T&b\end{pmatrix}\binom{X}{y}=X^TAX+2 y X^T U+by^2$$
in the way we would do it using numbers... only taking care of the respective dimensions.
Edit: an interesting transcription of a conference made by S. Wolfram here that I found referenced in a students' report.