Gaussian primes, quaternion primes, ... octonions?

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: $mathbb{C} times mathbb{H}^4$, ...