Here's the relationship between the $E_8$ Weyl group $W$ and the simple group $O_8^+(2)$, as I understand it. Some of this is contained in or follows from Daniel Allcock's "Ideals in the Integral Octaves" paper from 1998, and work of Conway et. al..

Let $\Omega$ be the $E_8$ lattice, which (up to scaling) I prefer to view as Coxeter's ring of integral octonions, with quadratic form given by the octonion norm $N: \Omega \rightarrow {\mathbb Z}$. Then $W$ acts on $\Omega$ by norm-preserving ${\mathbb Z}$-linear automorphisms. Reducing mod two, $W$ acts by ${\mathbb F}_2$-linear automorphisms on $\bar \Omega = \Omega / 2 \Omega$, preserving the reduction of the norm $\bar N : \bar \Omega \rightarrow {\mathbb F}_2$ (a quadratic form mod $2$).

This gives a homomorphism from $W$ to the orthogonal group $O(\bar \Omega, N)$, but not a homomorphism from $W$ to the special orthogonal group $SO(\bar \Omega, N)$; this is because the special orthogonal group is defined via the Dickson map rather than the determinant in characteristic two. The group $SO(\bar \Omega, N)$ is the finite simple group $O_8^+(2)$. As Noam commented, the subgroup $SO(\bar \Omega, N)$ can be defined as the subgroup of the orthogonal group $O(\Omega, N)$ consisting of elements fixing an even-dimensional subspace (one can realize the Dickson invariant here as the dimension of the fixed space, mod $2$). A simple Weyl reflection will fix a 7-dimensional subspace and 7 is odd :)

The moral is that there's not an interesting homomorphism from $W$ to $SO(\bar \Omega, N) = O_8^+(2)$.

But... if one takes the even subgroup $W^+ \subset W$, which coincides with the commutator subgroup $[W,W]$, then the image of $W^+$ in $O(\bar \Omega, N)$ coincides with $SO(\bar \Omega, N)$. The kernel of this homomorphism is the central subgroup $\{ \pm 1 \}$ in $W^+$. Thus there's a central extension, $$1 \rightarrow \{ \pm 1 \} \rightarrow W^+ \rightarrow SO(\bar \Omega, N) \rightarrow 1.$$

So to summarize, the Weyl group $W$ contains $W^+$ with index two, and the simple group $SO(\bar \Omega, N) = O_8^+(2)$ is a quotient of $W^+$ by a central subgroup of order two.

Here's the relationship between the $E_8$ Weyl group $W$ and the simple group that might be called $O_8^+(2)$ by some authors, as I understand it. Some of this is contained in or follows from Daniel Allcock's "Ideals in the Integral Octaves" paper from 1998, and work of Conway et. al..

Let $\Omega$ be the $E_8$ lattice, which (up to scaling) I prefer to view as Coxeter's ring of integral octonions, with quadratic form given by the octonion norm $N: \Omega \rightarrow {\mathbb Z}$. Then $W$ acts on $\Omega$ by norm-preserving ${\mathbb Z}$-linear automorphisms. Reducing mod two, $W$ acts by ${\mathbb F}_2$-linear automorphisms on $\bar \Omega = \Omega / 2 \Omega$, preserving the reduction of the norm $\bar N : \bar \Omega \rightarrow {\mathbb F}_2$ (a quadratic form mod $2$).

This gives a homomorphism from $W$ to the orthogonal group $O(\bar \Omega, N)$, but not a homomorphism from $W$ to the special orthogonal group $SO(\bar \Omega, N)$; this is because the special orthogonal group is defined via the Dickson map rather than the determinant in characteristic two. The group $SO(\bar \Omega, N)$ is a finite simple group which might be called $O_8^+(2)$ by some authors. Related to Noam's comment, the subgroup $SO(\bar \Omega, N)$ can be defined as the subgroup of the orthogonal group $O(\Omega, N)$ consisting of elements fixing an even-dimensional subspace (one can realize the Dickson invariant here as the dimension of the fixed space, mod $2$). A simple Weyl reflection will fix a 7-dimensional subspace and 7 is odd :)

The moral is that there's not an interesting homomorphism from $W$ to $SO(\bar \Omega, N)$.

But... if one takes the even subgroup $W^+ \subset W$, which coincides with the commutator subgroup $[W,W]$, then the image of $W^+$ in $O(\bar \Omega, N)$ coincides with $SO(\bar \Omega, N)$. The kernel of this homomorphism is the central subgroup $\{ \pm 1 \}$ in $W^+$. Thus there's a central extension, $$1 \rightarrow \{ \pm 1 \} \rightarrow W^+ \rightarrow SO(\bar \Omega, N) \rightarrow 1.$$

So to summarize, the Weyl group $W$ contains $W^+$ with index two, and the simple group $SO(\bar \Omega, N)$ is a quotient of $W^+$ by a central subgroup of order two.

Here's the relationship between the $E_8$ Weyl group $W$ and the simple group $O_8^+(2)$, as I understand it. Some of this is contained in or follows from Daniel Allcock's "Ideals in the Integral Octaves" paper from 1998, and work of Conway et. al..

Let $\Omega$ be the $E_8$ lattice, which (up to scaling) I prefer to view as Coxeter's ring of integral octonions, with quadratic form given by the octonion norm $N: \Omega \rightarrow {\mathbb Z}$. Then $W$ acts on $\Omega$ by norm-preserving ${\mathbb Z}$-linear automorphisms. Reducing mod two, $W$ acts by ${\mathbb F}_2$-linear automorphisms on $\bar \Omega = \Omega / 2 \Omega$, preserving the reduction of the norm $\bar N : \bar \Omega \rightarrow {\mathbb F}_2$ (a quadratic form mod $2$).

This gives a homomorphism from $W$ to the orthogonal group $O(\bar \Omega, N)$, but not a homomorphism from $W$ to the special orthogonal group $SO(\bar \Omega, N)$; this is because the special orthogonal group is defined via the Dickson map rather than the determinant in characteristic two. The group $SO(\bar \Omega, N)$ is the finite simple group $O_8^+(2)$. As Noam commented, the subgroup $SO(\bar \Omega, N)$ can be defined as the subgroup of the orthogonal group $O(\Omega, N)$ fixing an even-dimensional subspace (one can realize the Dickson invariant here as the dimension of the fixed space, mod $2$). A simple Weyl reflection will fix a 7-dimensional subspace and 7 is odd :)

The moral is that there's not an interesting homomorphism from $W$ to $SO(\bar \Omega, N) = O_8^+(2)$.

But... if one takes the even subgroup $W^+ \subset W$, which coincides with the commutator subgroup $[W,W]$, then the image of $W^+$ in $O(\bar \Omega, N)$ coincides with $SO(\bar \Omega, N)$. The kernel of this homomorphism is the central subgroup $\{ \pm 1 \}$ in $W^+$. Thus there's a central extension, $$1 \rightarrow \{ \pm 1 \} \rightarrow W^+ \rightarrow SO(\bar \Omega, N) \rightarrow 1.$$

So to summarize, the Weyl group $W$ contains $W^+$ with index two, and the simple group $SO(\bar \Omega, N) = O_8^+(2)$ is a quotient of $W^+$ by a central subgroup of order two.

Here's the relationship between the $E_8$ Weyl group $W$ and the simple group $O_8^+(2)$, as I understand it. Some of this is contained in or follows from Daniel Allcock's "Ideals in the Integral Octaves" paper from 1998, and work of Conway et. al..

Let $\Omega$ be the $E_8$ lattice, which (up to scaling) I prefer to view as Coxeter's ring of integral octonions, with quadratic form given by the octonion norm $N: \Omega \rightarrow {\mathbb Z}$. Then $W$ acts on $\Omega$ by norm-preserving ${\mathbb Z}$-linear automorphisms. Reducing mod two, $W$ acts by ${\mathbb F}_2$-linear automorphisms on $\bar \Omega = \Omega / 2 \Omega$, preserving the reduction of the norm $\bar N : \bar \Omega \rightarrow {\mathbb F}_2$ (a quadratic form mod $2$).

This gives a homomorphism from $W$ to the orthogonal group $O(\bar \Omega, N)$, but not a homomorphism from $W$ to the special orthogonal group $SO(\bar \Omega, N)$; this is because the special orthogonal group is defined via the Dickson map rather than the determinant in characteristic two. The group $SO(\bar \Omega, N)$ is the finite simple group $O_8^+(2)$. As Noam commented, the subgroup $SO(\bar \Omega, N)$ can be defined as the subgroup of the orthogonal group $O(\Omega, N)$ consisting of elements fixing an even-dimensional subspace (one can realize the Dickson invariant here as the dimension of the fixed space, mod $2$). A simple Weyl reflection will fix a 7-dimensional subspace and 7 is odd :)

The moral is that there's not an interesting homomorphism from $W$ to $SO(\bar \Omega, N) = O_8^+(2)$.

But... if one takes the even subgroup $W^+ \subset W$, which coincides with the commutator subgroup $[W,W]$, then the image of $W^+$ in $O(\bar \Omega, N)$ coincides with $SO(\bar \Omega, N)$. The kernel of this homomorphism is the central subgroup $\{ \pm 1 \}$ in $W^+$. Thus there's a central extension, $$1 \rightarrow \{ \pm 1 \} \rightarrow W^+ \rightarrow SO(\bar \Omega, N) \rightarrow 1.$$

So to summarize, the Weyl group $W$ contains $W^+$ with index two, and the simple group $SO(\bar \Omega, N) = O_8^+(2)$ is a quotient of $W^+$ by a central subgroup of order two.