The correct options are:

Option BB and Option DD.

These relations are based on the properties of the imaginary units $i, j, k$ in the context of quaternion algebra. Specifically:

1. Option BB: $i \times j = -k, j \times i = k$

   • This is true and aligns with the rules of quaternion multiplication.

2. Option DD: $k \times i = -j, i \times k = j$

   • This is also true and consistent with quaternion multiplication properties.

Options AA and CC are incorrect because:

• Option AA: $i \times i = j \times j = k \times k = 1$
  • In quaternion algebra, $i \times i = j \times j = k \times k = -1$, not 1.
• Option CC: $i \times i = j \times j = k \times k = 0$
  • Similarly, this does not hold true in quaternion algebra, as mentioned above.

Do you want details or have any questions?

Here are 5 numbered questions for you to ask:

1. What are quaternions and how do they work?
2. Can you explain the basic properties of quaternion multiplication?
3. How do quaternions differ from complex numbers?
4. What are some practical applications of quaternions?
5. Can you show a step-by-step example of quaternion multiplication?

Tip: When dealing with quaternions, always remember the key multiplication rules for the imaginary units $i, j,$ and $k$ to avoid mistakes.