Provide option for dollar sign to be treated as bracket

[crxyz0]

I would love a feature so that when you type a dollar sign a second is automatically created, just like when you type any other bracket. And, like brackets, when you type a $ when your cursor is directly to the right of the closing dollar sign, your cursor moves one to the right without creating another dollar sign. Currently, when you type a dollar sign, only one appears. This feature would be very convenient for typing math equations.

[octaeder]

out of the box you have Ctrl+Shift+M for inline math environments (s. menu Math).
you may also want to look into chapter Personal Macros in Advanced features of the manual.