This is not a precise question. I am curious to know how do you present to your students the (imprecise) concept of "without loss of generality", and how to use it correctly/incorrectly.
I got curious after a student of mine (in calculus $I$), while trying to prove that for every $a \in \mathbb{R}$ $$ \big(\frac{a+|a|}{2}\big)^2+\big(\frac{a-|a|}{2}\big)^2=a^2, $$ wrote that without loss of generality, we may assume that $a \ge 0$.
I decided not to accept this usage of the concept as legitimate. Both sides of the equations are invariant under the transformation $a \to -a$, so one can appeal to symmetry here, but this appealing to symmetry needs to be stated (especially since it is not immediately obvious for the LHS).
Anyway, this got me thinking about how to explain to students the slippery concept of WLOG.
I think that a good way is using examples, so I am asking for elementary examples (suitable for first semester math courses) which illustrate the concept. "Anti-examples" where this usage is less appropriate (in a non-trivial way) will also be nice, in order to warn the students from making mistakes.
To start the ball rolling, here is a classic example:
We want to prove some property $P(x,y)$ holds for every real numbers $x,y$, and $P$ is symmetric, i.e. $P(x,y) \iff P(y,x)$.
Let $Q(x,y)$ be some "property" (logical formula) satisfying $$\mathbb{R}^2=\{ (x,y) \in \mathbb{R}^2 \, | \, Q(x,y) \} \cup \{ (x,y)\in \mathbb{R}^2 \, | \, Q(y,x) \}. \tag{1}$$
Then we can assume W.L.O.G that $Q(x,y)$ holds, i.e. it suffices to prove that $Q(x,y) \Rightarrow P(x,y)$:
Indeed, $Q(y,x) \Rightarrow P(y,x) \Rightarrow P(x,y)$, and by our assumption $(1)$, this justifies the reduction. (A common choice for $Q$ is $Q(x,y):=x \le y$).
Here is a summary of an informal introduction to the concept:
The term W.L.O.G is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic, or follows by a symmetry argument.
Ultimately, it is intentionally introducing an easy-to-fill gap in a proof and relying on the reader to fill the gap.