First, suppose $\mathcal{F}$ is a frame validating 4, with $wRu$ and $uRt$. Let $M$ be a model with $V(p)=\{v\mid wRv\}$, where $p$ a propositional variable. So $\models_{w}\square p$. By assumption, we have $\models_{w}\square p\to\square\square p$. Then $\models_{w}\square\square p$. This means $\models_{v}\square p$ for all $v$ such that $wRv$. Since $wRu$, $\models_{u}\square p$, which means $\models_{s}p$ for all $s$ such that $uRs$. Since $uRt$, we have $\models_{t}p$, or $t\in V(p)$, or $wRt$. Hence $R$ is transitive.

Conversely, let $\mathcal{F}$ be a transitive frame, $M$ a model based on $\mathcal{F}$, and $w$ any world in $M$. Suppose $\models_{w}\square A$. We want to show $\models_{w}\square\square A$, or for all $u$ with $wRu$, we have $\models_{u}\square A$, or for all $u$ with $wRu$ and all $t$ with $uRt$, we have $\models_{t}A$. If $wRu$ and $uRt$, $wRt$ since $R$ is transitive. Then $\models_{t}A$ by assumption. Therefore, $\models_{w}\square A\to\square\square A$. ∎

Since any theorem in S4 is deducible from a finite sequence consisting of tautologies, which are valid in any frame, instances of T, which are valid in reflexive frames, instances of 4, which are valid in transitive frames by the proposition above, and applications of modus ponens and necessitation, both of which preserve validity in any frame, whence the result. ∎

Since S4 contains T, its canonical frame $\mathcal{F}_{{\textbf{S4}}}$ is reflexive. We next show that the canonical frame $\mathcal{F}_{{\Lambda}}$ of any consistent normal logic $\Lambda$ containing the schema 4 must be transitive. So suppose $wR_{{\Lambda}}u$ and $uR_{{\Lambda}}v$. If $A\in\Delta_{w}:=\{B\mid\square B\in w\}$, then $\square A\in w$, or $\square\square A\in w$ by modus ponens on 4 and the fact that $w$ is closed under modus ponens. Hence $\square A\in\Delta_{w}$, or $\square A\in u$ since $wR_{{\Lambda}}u$, or $A\in\Delta_{u}$, or $A\in v$ since $uR_{{\Lambda}}v$. As a result, $wR_{{\Lambda}}v$, and therefore $\mathcal{F}_{{\textbf{S4}}}$ is a preordered frame. ∎