From Wikipedia, a graded Lie algebra is defined as a direct sum of vector spaces $$ \mathfrak{g} = \bigoplus_{i \in \mathbb{Z}} \mathfrak{g}_i \tag{1} $$ such that the Lie bracket satisfies $$ [\mathfrak{g}_i, \mathfrak{g}_j] \subseteq \mathfrak{g}_{i+j} \tag{2} $$
So from a physicists point of view, I interpret equation (2) as follows:
The Lie bracket of an element $x \in \mathfrak{g}_i$ and an element $y \in \mathfrak{g}_j$ will be an element in the vector space $\mathfrak{g}_{i+j}$. But let us now, for definiteness, assume the graded Lie algebra is given by
$$
\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2 \oplus \mathfrak{g}_3
$$
Then for $x \in \mathfrak{g}_2$ and $y \in \mathfrak{g}_3$
$$
[x, y] \subseteq \mathfrak{g}_{2+3} = \mathfrak{g}_5
$$
But what is $\mathfrak{g}_5$ in this example (considering that we have only assumed the existence of $\mathfrak{g}_1$, $\mathfrak{g}_2$ and $\mathfrak{g}_3$)?
Here ${\cal g}_i$ will be zero for $i>3$ so if $x\in {\cal g}_2, y\in {\cal g}_3, [x,y]=0$.