English
For any quaternion a, the square of its imaginary part equals the negative of the norm of that imaginary part: (a.im)^2 = - normSq(a.im).
Русский
Для любого квартониона a квадрат мнимой части равен минус квадрат нормы этой мнимой части: (a.im)^2 = -normSq(a.im).
LaTeX
$$$(a.im)^2 = -\operatorname{normSq}(a.im)$$$
Lean4
theorem im_sq : a.im ^ 2 = -normSq a.im := by simp_rw [sq, ← star_mul_self, star_im, neg_mul, neg_neg]