Introduction: Solving the "Bracket System".
Now , let's use the same logic in a Lie Algebra context. In Lie Theory , we often need to "decompose" the result of a bracket into its basic components. This is exactly like solving a system of equations.

Exercise: In `sl(2 , RR)` , we have a basis `{e , f , h}`. Suppose we are told that the bracket of two unknown vectors `v` and `w` results in the matrix:

`K = ([5 , 3], [2 , -5])` (the bracket of `v` and `w`)

The Task: Find the unique coordinates `(c_1 , c_2 , c_3)` such that `K = c_1 e + c_2 f + c_3 h`.

General Strategy: We set up a system of linear equations by matching the entries of the matrix `K` with the entries of the basis matrices `e , f` and `h`.

Specific Theory Used:

- Basic definitions: `e = ([0 , 1] , [0 , 0]) , f = ([0 , 0] , [1 , 0]) , h = ([1 , 0] , [0 , -1])`

- Matrix Addition: `c_1 e + c_2 f + c_3 h = ([c_3 , c_1] , [c_2 , -c_3])`

Solution: To solve ` K = c_1 e + c_2 f + c_3 h` , we write:

`([5 , 3] , [2 , -5]) = ([c_3 , c_1] , [c_2 , .c_3])`

This gives us a system of equations by comparing each slot:
1. Top-left: `c_3 = 5`
2. Top-right: `c_1 = 3`
3. Botton-left: `c_2 = 2`
4. Bottom-right: `-c_3 = -5` (This confirms 1. , our first equation)

Result: The solution is ` K = 3e + 2f + 5h`. Because `{e , f , h}` is a basis , we know this is the only way to represent the result of that bracket. QED