Subalgebras, quotients, and extensions of solvable Lie algebras

Induction on $r$ gives ${\mathfrak{h}}^{(r)}\subseteq {\mathfrak{g}}^{(r)}$: it is clear at $r=0$, and bracketing a subspace with itself preserves containment. Therefore a vanishing derived term of $\mathfrak{g}$ forces the corresponding term of $\mathfrak{h}$ to vanish.

Bracket preservation and surjectivity of the canonical map in [L2] give $(\mathfrak{g}\mathrm{/}\mathfrak{i}{)}^{(r)}=\pi ({\mathfrak{g}}^{(r)})$ for every $r$. Thus every solvable quotient of $\mathfrak{g}$ terminates no later than $\mathfrak{g}$; [L3] records the same calculation for any homomorphic image.

Suppose $m=\mathrm{dl}(\mathfrak{g}\mathrm{/}\mathfrak{i})$ and $n=\mathrm{dl}(\mathfrak{i})$. Step 1.2 gives $\pi ({\mathfrak{g}}^{(m)})=0$, so ${\mathfrak{g}}^{(m)}\subseteq \ker \pi =\mathfrak{i}$. Iterating the derived operation $n$ further times gives ${\mathfrak{g}}^{(m+n)}=({\mathfrak{g}}^{(m)}{)}^{(n)}\subseteq {\mathfrak{i}}^{(n)}=0$. This proves solvability and the bound, including $m=0$ or $n=0$.