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Basic Introduction

Given two whole numbers, e.g. 791 and 636, two questions to ask first are:

1. Are these floretions?
2. How is multiplication defined?

A formal definition is given here A308496. Assuming the reader is familiar with quaternion multiplication, here is a short answer:

Steps

1. Convert each number to octal: 791 → 1427, 636 → 1174
2. Check that all digits in octal are from the set S= {1, 2, 4, 7}. If not, these are not floretions.
3. Check that the order of each floretion is the same. The order is the number of digits in octal. Both 791 and 636 have 4 octal digits. If they are not the same order, they are members of two separate groups and cannot be multiplied together.
4. Associate digits with quaternions as: 1 ↔ i, 2 ↔ j, 4 ↔ k, 7 ↔ e where the latter is the unit. Thus 1427 becomes ikje and 1174 becomes iiek
5. Multiply ikje · iiek quaternion by quaternion, bringing any negative signs out the front:

ikje · iiek = (i · i)(k · i)(j · e)(e · k) = (-e)(j)(j)(k) = -ejjk