Alright I'm going to try one last time to explain my problem with quaternions and multiplication of two quaternions in specific. This time hopefully I'll get an explanation that makes sense. (I posted 2 other versions of this question before that I wasn't happy with so I scrapped them)
Assume we have a quaternion $q \in \Bbb{H}$, where $q = w + xi + yj + zk$ with $w, x, y, z \in \Bbb{R}$. We can represent this quaternion in terms of an ordered pair $[s, v]$ where $s$ is the scalar component (i.e. $s = w$) and $v$ is a vector in $\Bbb{R}^3$ such that $v = xi + yj + zk$.
Now, moving on to the derivation of multiplication of two quaternions. Assuming we had $q_a = [s_a, a]$ and $q_b = [s_b, b]$.
$$ q_aq_b = [s_a, a][s_b, b] = (s_a + x_ai + y_aj + z_ak)(s_b + x_bi + y_bj + z_bk) \\ = (s_as_b + s_ax_bi + s_ay_bj + s_az_bk) + (s_bx_ai - x_ax_b + x_ay_bk - x_az_bj) + (s_by_aj - y_ax_bk - y_ay_b + y_az_bi) + (s_bz_ak + z_ax_bj - z_ab_yi - z_az_b)$$
$$ = (s_as_b - x_ax_b - y_ay_b - z_az_b) + (s_ax_b + s_bx_a + y_az_b - z_ay_b)i + (s_ay_b - x_az_b + s_by_a + z_ax_b)j + (s_az_b + x_ay_b - y_ax_b + s_bz_a)k $$
$$ = (s_as_b - x_ax_b - y_ay_b - z_az_b) + s_a(x_bi + y_bj + z_bk) + s_b(x_zi + y_aj + z_ak) + (y_az_b - z_ay_b)i + (z_ax_b - x_az_b)j + (x_ay_b - y_ax_b)k $$ $$ = [s_as_b - x_ax_b - y_ay_b - z_az_b, s_a(x_bi + y_bj + z_bk) + s_b(x_zi + y_aj + z_ak) + (y_az_b - z_ay_b)i + (z_ax_b - x_az_b)j + (x_ay_b - y_ax_b)k] $$ $$ = [s_as_b - x_ax_b - y_ay_b - z_az_b, s_ab + s_ba + a \times b] $$ Where $a \times b$ denotes the cross product between $a$ and $b$. So far so good, maybe a little messy but we're getting somewhere. The thing that confuses me now is simplification of the scalar component of the result ordered pair (which is $s_as_b - x_ax_b - y_ay_b - z_az_b$).
One can observe that this is incredibly close to the dot product of $a$ and $b$: $$ a \cdot b = x_ax_bi^2 + y_ay_bj^2 + z_az_bk^2 \\ = -x_ax_b - y_ay_b - z_az_b $$ This is because $$ i^2 = j^2 = k^2 = ijk = -1 $$ Thus, the scalar component of the ordered pair should be equal to $s_as_b + a \cdot b$, right? WRONG
According to here, here, here, and any other resource you can find, the scalar component of $q_aq_b$ would ACTUALLY be $s_as_b - a \cdot b$. This doesn't make a lick of sense to me. All the terms in $a \cdot b$ are already negative, so negating them again would only make them positive, giving you $s_as_b + x_ax_b + y_ay_b + z_az_b$, which is not what the calculations above revealed. What am I doing wrong? What am I missing? It feels like it's such an obvious step that I'm missing but I just can't figure it out.