naturally graded algebras I found a good explanation on graded algebras Understanding of graded algebra But I am confused about difference between " **naturally graded** " and " **graded** ". May you please clarify the notion of naturally graded with an example. Let consider Lie algebra $sl_2$ over field of characteristic $0$. We have, for example, $\mathbb{Z}$, ${\mathbb{Z}}_{2} \times {\mathbb{Z}}_{2} $ and ${\mathbb{Z}}_{2} $ grading for that. Do mentioned grading are naturally grading? Does a naturally graded mean $\mathbb{N}$-grading?

"Naturally graded" is not a technical term; it is entirely informal and has no precise meaning. It just means that the grading "arises naturally" rather than being some artificial ad hoc grading that someone made up.