I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as
A transformation which preserves the structure of space
What does he mean by the structure of space? Why can reflection preserve the structure?
On
The word automorphism can be used in a lot of contexts, as for example the sense here of a mapping of space (?) onto itself in a way that "preserves the structure".
This is a reference to the Euclidean properties of space, i.e. its distances and angles. Reflection in any plane preserves the measures of distance and angles, and so is an automorphism in that sense.
On
https://en.wikipedia.org/wiki/Weyl_group
I am pretty sure he is discussing, without using the terms, Lie Algebras and what came to be known as Weyl Chambers.
The setting for a root lattice is a (positive) quadratic form, let us call it $Q.$ The (squared) norm of a lattice vector $x$ is precisely $Q(x)$ which is an integer; for the cases of interest, this is usually going to be required to be a even integer, as the root lattices are "even."
The quadratic form gives rise to a symmetric inner product, $$ \langle x,y \rangle = (1/2) \left( Q(x+y) - Q(x) - Q(y) \right). $$ It is required that this be an integer when $x,y$ are in the lattice. This is the reason that norms must come out even, the whole business needs to be doubled to avoid half-integers.
Given an inner product, the reflection in a fixed vector $w$ is $$ z' = z - \frac{2 \langle z,w \rangle}{\langle w,w \rangle} w $$ You have seen this before, probably with ordinary dot product. Note that $w' = -w,$ but when $z$ is perpendicular to $w,$ then $z'=z.$
Next, given a even lattice, a root is a lattice vector $r$ such that $Q(r) = 2,$ or $\langle r,r \rangle = 2$. It is because of this condition that reflection in $r$ becomes $$ \color{magenta}{ z' = z - \langle z,r \rangle r} $$ Given any $z$ in the lattice, $z'$ is also in the lattice. This preserves the lattice and is called an automorphism. With positive $Q,$ the automorphism group is finite and is generated by the reflections.
Well, there is more to be said. I am now working in quadratic forms. I feel that if I had had a short course in quadratic forms, lattices, before Lie Algebras, it would have helped.
Enough for now
The phrase "structure of a space" has different meaning in different contexts.
The quote from Weyl probably gives the structure implicitly, and I am guessing that he is talking about planes in Euclidean 3-space. In that case, an automorphism is a bijection of the points of Euclidean 3-space which preserves all the structures of Euclidean geometry: lines; planes; distance; angle. In analytic geometry we learn a powerful theorem which says that any bijection of the points of Euclidean 3-space that preserves distance also preserves all the other structure: lines; planes; angle.
So Weyl's quote simply means that reflection across a plane preserves distance: using Cartesian coordinates $\mathbb{R}^3$ for Euclidean 3-space, with the usual distance formula $d(x,y)$ for points $x,y \in \mathbb{R}^3$, if $P \subset \mathbb{R}^3$ is the plane and if $r : \mathbb{R}^3 \to \mathbb{R}^3$ is the reflection across $P$ then $d(r(x),r(y)) = d(x,y)$. This is pretty easy to prove using ordinary tools of analytic geometry: use a description of $P$ to write down a formula for $r$ and use that formula with coordinates for $x,y$ to prove $d(r(x),r(y))=d(x,y)$.