I came across the following example of a restricted Lie algebra in the book Frobenius Algebras I by Skowroński and Yamagata.
Given a field $K$ of characteristic $p > 0$, a restricted Lie algebra over $K$ is a triple $(L, [-, -], {}^{[p]})$ where $(L, [-, -])$ is a Lie algebra over $K$ and ${}^{[p]}$ is a (set-theoretic) map $L \to L, x \mapsto x^{[p]}$ satisfying certain conditions (See page 580 of the above book, Wikipedia, or this question on MSE for details).
The first example (Example 2.21 (a)) of a restricted Lie algebra over $K$ is constructed as follows:
Let $L = K^2$ and $e_1 = (1, 0)$, $e_2 = (0, 1)$ be the canonical basis of $L$ over $K$. Consider the $K$-bilinear map $[-, -] \colon L \times L \to L$ given by $$[e_1, e_1] = 0, \quad [e_2, e_2] = 0, \quad [e_1, e_2] = e_2, \quad [e_2, e_1] = -e_2,$$ and the map ${}^{[p]} \colon L \to L$ given by $x^{[p]} = \lambda_1^pe_1$ for $x = \lambda_1e_1 + \lambda_2e_2 \in L$.
I doubt this defines a restricted Lie algebra. One property that the triple $(L, [-, -], {}^{[p]})$ must satisfy (except for $(L, [-, -])$ being a Lie algebra) is that $\operatorname{ad}{(x^{[p]})} = (\operatorname{ad}{x}) ^ p$ for all $x \in L$, where for $z \in L$, $\operatorname{ad}{z} \, \colon L \to L$ denotes the morphism $y \mapsto [z, y]$.
However, for example when $p = 3$, we have
$$\operatorname{ad}((e_1 + e_2)^{[3]})(e_1) = \operatorname{ad}(e_1)(e_1) = [e_1, e_1] = 0.$$
Meanwhile,
$$\begin{align}
\operatorname{ad}(e_1 + e_2)(e_1) & = [e_1 + e_2, e_1] = [e_1, e_1] + [e_2, e_1] = -e_2; \\
\operatorname{ad}(e_1 + e_2)(e_2) & = [e_1 + e_2, e_2] = [e_1, e_2] + [e_2, e_2] = e_2,
\end{align}$$
gives
$$\operatorname{ad}(e_1 + e_2)^3(e_1)
= -\operatorname{ad}(e_1 + e_2)^2(e_2)
= -\operatorname{ad}(e_1 + e_2)(e_2)
= -e_2.$$
Hence $\operatorname{ad}((e_1 + e_2)^{[3]}) \ne \operatorname{ad}(e_1 + e_2)^3$.
Are there something wrong in my calculation? I would also like to know if slight modification turns this (seemingly non-)example into an example of a restricted Lie algebra.