Say you’ve got a document with a bunch of exercises and a few lemmas. You may want the exercises numbered separately from the lemmas and theorems, unlike LyX’s default behavior. This can be achieved by redefining xca, the environment LyX uses for exercises. Add the following to your LaTeX preamble:

letxca@undefined
theoremstyle{plain}
newtheorem{xca}{protectexercisename}

LyX will still display the incorrect numbering, but the output will be correct nonetheless. The first line undefines LyX’s definition of xca, then we set the style to match the old one, and we redefine xca, this time without a reference to the theorem counter.

After looking for a built-in expectation symbol in LaTeX, and coming up with none, I’ve defined one. Just add:

% Expectation symbol
DeclareMathOperator*{E}{mathbb{E}}

to your LaTeX preamble, and you’re done. You’ll also need to add usepackage{amsmath}, or in LyX, tick “Use AMS math package” under Document->Settings->Math Options.

Using the starred version of DeclareMathOperator makes sure subscripts go beneath the symbol in display mode.

I believe LyX is a great tool for writing LaTeX documents. It makes writing formulas very easy, and it allows you to see the formula as you are writing, as opposed to seeing only LaTeX code. However, LyX doesn’t support every LaTeX package and the macros it defines. Sure, it doesn’t stop you from using these macros in your formulas, but they don’t display nicely; you see the name of the macro instead of a schematic preview.

While LyX doesn’t support many of the great packages out there, like mathtools (which I really hope it will someday), you can add some support to your documents. At the beginning of the document, insert a comment via Insert->Note->Comment. Inside the newly created comment, insert a math macro via Insert->Math->Macro. In the name part, put the name of the command you want to add support for. In the second box (captioned LyX), use existing LyX commands to mimic how the macro will look. For example, this is what it looks like for the coloneqq macro (from the great mathtools package):

After adding the math macro in the comment, when you use the macro inside formulas, it will display nicely:

A little explanation of how things work: when you define a math macro in LyX, LyX does two things:

1. Inserts LaTeX code to create the macro.
2. Displays the macro nicely when editing the document.

While the latter is desirable, the former is problematic. If LyX inserts LaTeX code to define the existing macro, it will cause errors. So when you put the LyX macro in the comment environment, the code LyX generates gets ignored, and only the second, desirable, outcome is achieved.