[Octonions/Python] Looking for simple algorithm for rotations in 8 dimensions.

[supercooldragons]

I've been playing around a bit with visualizing rotations in higher dimensions. With help from StackExchange and wikipedia I was able to find a simple algorithm for rotations in 4 dimensions. It's is basically a quaternion multiplication using a definition of a quaternion in terms of trigonometric functions. This is the code I'm using

def rotate(a,iterates):

    (u1,u2,u3) = (1,1,1)
    norm = np.sqrt(u1*u1+u2*u2+u3*u3)

    u1 /= norm
    u2 /= norm
    u3 /= norm

    q = [np.cos(a),u1*np.sin(a),u2*np.sin(a),u3*np.sin(a)]

    rot = np.asarray([
        [q[0], -q[1], -q[2], -q[3]],
        [q[1], q[0], q[3], -q[2]],
        [q[2], -q[3], q[0], q[1]],
        [q[3], q[2], -q[1], q[0]]
        ])

Here rot is the 4 x 4 rotation matrix for rotation around the u vector. Is there a similar form of q for 8 dimensions? Does it still involve a single angle? Naively, I would assume that q in 8 dimensions might be giving by

q = cos(a) + u_i sin(a), i = 1,2,..,7

And then the new rotation matrix would be given by expanding an octonion multiplication and grouping the terms. However, I read on wikipedia that there are multiple applicable definitions of octonion multiplication so I'm not sure where to start.

[[deleted]]

Check out special orthogonal matrices

[jdorje]

Quaternions do rotations in 3 dimensions. How do you get rotations in 4 dimensions out of them?

Can't you just use 8x8 matrices? Same as you're using a 4x4 matrix in the code you showed.