I got stucked in the definition of "symmetry" in the chapter of Lie Algebras to understand later the root systems. Well in the script they used the following definition:
Let $\alpha \in V\setminus \{0\}$. A symmetry with vector $\alpha$ is an element of $s \in GL(V)$ with $$s(v)=v-\alpha^*(v)\alpha$$
for all $v \in V$ with $\alpha^*(\alpha)=2$.
Now the book of Serre "Complex Semisimple Lie Algebra" gives us an other definition:
Let $\alpha \in V\setminus \{0\}$. One defines a symmetry with vector $\alpha$ to be any automorphism $s$ of $V$ satisfying the following two conditions:
(i) $s(\alpha) = - \alpha$
(ii) The set $H$ of elements of $V$ fixed by $s$ is a hyperplane of $V$.
I know don't see the relation between these two definitions. Especially the second point in the second definition confuses me a lot. Also what can I understand under the expression $\alpha^*(\alpha)$?
Many thanks for some help.
Both of these definitions capture the idea of "reflection in the hyperplane perpendicular to $\alpha$". Concretely, if $v \in V$ you can write $v = a\alpha + h$ where $h \perp \alpha$, and this reflection sends $v$ to $-a\alpha + h$. (In particular, it fixes everything in the hyperplane perpendicular to $\alpha$).
$\alpha^*$ refers to the linear map $V \to \mathbb{R}$ (or whatever field $V$ is a vector space over) sending $\alpha$ to 2 and sending everything perpendicular to $\alpha$ to 0, so $\ker \alpha^*$ is the hyperplane perpendicular to $\alpha$. The notation $\alpha^*(\alpha)$ just means evaluate the function $\alpha^*:V \to \mathbb{R}$ at $\alpha \in V$.
If $\alpha^*$ is such a map, you can check that $s(v) = v - \alpha^*(v)a$ really sends $a\alpha + h$ to $-a\alpha +h$ so it is the reflection described above.
In (ii), "The set $H$ of elements fixed by $s$" means $H=\{v \in V: s(v)=v\}$, and saying that this is a hyperplane means that $\dim H = \dim V - 1$.