Is there a notion of an octonion prime?
A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime.
A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is prime.
I know there is an eight-square identity that underlies the octonions.
Is there a parallel statement, something like: an octonion is prime if its norm is prime?


I ask out of curiosity and ignorance.


**Addendum.**
The Conway-Smith book Bruce recommended is a great source on my question.
As there are several candidates for what constitutes an integral octonion,
the situation is complicated. But a short answer is that unique factorization
fails to hold, and so there is no clean notion of an octonion prime.
C.-S. select out and concentrate on what they dub the *octavian integers*, which,
as Bruce mentions, geometrically form the $E^8$ lattice.
Here is one pleasing result (p.113): If $\alpha \beta = \alpha' \beta'$, where
$\alpha, \alpha', \beta, \beta'$ are nonzero octavian integers, then the angle between $\alpha$ and $\alpha'$ is
the same as the angle between $\beta$ and $\beta'$.




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A non-serious postscript:
Isn't it curious that 
$\mathbb{N}$,
$\mathbb{C}$,
$\mathbb{H}$,
$\mathbb{O}$
correspond to N, C, H, O, the four atomic elements that comprise all proteins and much of organic
life? Water-space: $\mathbb{H}^2 \times \mathbb{O}$, methane-space: $\mathbb{C} \times \mathbb{H}^4$, ...



Gaussian primes, quaternion primes, ... octonions?
You should probably read "On Quaternions and Octonions" by J.H. Conway and D.A. Smith


P.S. It's "octonion" not "octonian"


Edit: The first thing you will find is a discussion of integral numbers. For the complex numbers you have $\mathbb{Z}[i]$ (aka Gaussian integers) which is an $A\_1\times A\_1$ lattice. You also have $\mathbb{Z}[\omega]$ (aka Eisenstein integers) which is an $A\_2$-lattice. For quaternions you have the integral quaternions (as above) which is an $A\_1^4$ lattice and you also have the Hurwitz integral quaternions (adjoin $(1+i+j+k)/2$) which as a lattice is $D\_4$. The Hurwitz numbers have "division with small remainder" property which makes them better. For the octonions you might start with doubling the Hurwitz integers to get a $D\_4\times D\_4$ lattice. Then you might add more and get $D\_8$. However the octavian integers are an $E\_8$ lattice. They are not unique (there are seven versions). These have several good properties:


Every left or right ideal is principal.  

Every ideal is two-sided.


Then there is a discussion of prime factorisations.


Finally the automorphism group of the octaves has a simple subgroup of index 2 and order 12096. This group is $G\_2(2)$.


