Let $L$ be a Lie algebra over field $F$, $I$ - ideal in this algebra. It's stated that $I^2$ (and so any item of central series) is also ideal in $L$.
1) For any $a, b \in I^2: [a,b] \in I^2$. True, since $a, b \in I$.
2) For any $k \in F, a \in I^2: ka \in I^2$. True, because $a = [a_1,a_2],$ $a_i \in I$, and $ka = k[a_1,a_2] = [ka_1,a_2]$. As $ka_1, a_2$ are from $I$, so $[ka_1,a_2]$ is from $I^2$.
3) For any $a, b \in I^2: a + b \in I^2$. Why is that? How could you represent $[a_1,a_2] + [b_1,b_2]$ as $[c_1, c_2]$ for some $c_1,c_2 \in I$?
Let $I,J$ be two ideals of a Lie algebra $L$. The subspace $[I,J]$ generated by all brackets $[i,j]$ with $i\in I$ and $j\in J$ is an ideal of $L$. Indeed, for $x\in L$ and $i\in I,j\in J$, $$[x,[i,j]]=[\underbrace{[x,i]}_{\in I},j]+[i,\underbrace{[x,j]}_{\in J}]\in[I,J]$$ since $I$ and $J$ are ideals. In particular, $I^2=[I,I]$, the subspace of $L$ generated by all $[i,i']$, with $i,i'\in I$, is an ideal of $L$.