四元数の行列表現において，可逆元とその逆元を左右から作用させる
$$ \begin{eqnarray} \mathbf{q}=w_q+x_qi+y_qj+z_qk &\leftrightarrow& \href{https://shikitenkai.blogspot.com/2020/06/blog-post_99.html}{ \begin{bmatrix} w_q+x_qi&y_q+z_qi\\ -(y_q-z_qi)&w_q-x_qi\\ \end{bmatrix} } \\ \mathbf{p}=w_p+x_pi+y_pj+z_pk &\leftrightarrow& \begin{bmatrix} w_p+x_pi&y_p+z_pi\\ -(y_p-z_pi)&w_p-x_pi\\ \end{bmatrix} \end{eqnarray} $$ $$ \begin{eqnarray} \mathbf{q}^{-1}=\frac{\bar{\mathbf{q}}}{|\mathbf{q}|^2} &\leftrightarrow& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \end{eqnarray} $$ $$ \begin{eqnarray} \mathbf{q}\mathbf{p}\mathbf{q}^{-1} &\leftrightarrow& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} w_q+x_qi&y_q+z_qi\\ -(y_q-z_qi)&w_q-x_qi\\ \end{bmatrix} \begin{bmatrix} w_p+x_pi&y_p+z_pi\\ -(y_p-z_pi)&w_p-x_pi\\ \end{bmatrix} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} (w_q^2+x_q^2+y_q^2+z_q^2) w_p +( (w_q^2+x_q^2-y_q^2-z_q^2) x_p +2(x_qy_q-w_qz_q) y_p +2(w_qy_q+x_qz_q) z_p )i & 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p +( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i \\ -( 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p -( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i ) & (w_q^2+x_q^2+y_q^2+z_q^2)w_p -( (w_p^2+x_q^2-y_q^2-z_q^2)x_p+2(x_qy_q-w_qz_q)y_p+2(w_qy_q+x_qz_q)z_p )i \\ \end{bmatrix} \\&&\;\cdots\;式展開は最後に添付 \end{eqnarray} $$ $$ \begin{eqnarray} w_{qpq^{-1}}&=&\frac{(w_q^2+x_q^2+y_q^2+z_q^2) w_p}{w_q^2+x_q^2+y_q^2+z_q^2}=w_p \\x_{qpq^{-1}}&=&\frac{(w_q^2+x_q^2-y_q^2-z_q^2)x_p+2(x_qy_q-w_qz_q)y_p+2(w_qy_q+x_qz_q)z_p}{w_q^2+x_q^2+y_q^2+z_q^2} \\y_{qpq^{-1}}&=&\frac{2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p}{w_q^2+x_q^2+y_q^2+z_q^2} \\z_{qpq^{-1}}&=&\frac{2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p}{w_q^2+x_q^2+y_q^2+z_q^2} \end{eqnarray} $$特殊なp, qを考える
\( w_q=\cos{\left(\frac{\theta}{2}\right)},x_q^2+y_q^2+z_q^2=\sin^2{\left(\frac{\theta}{2}\right)} \)となる \(\mathbf{q}\)及び \( w_p=0 \)となる\(\mathbf{p}\)を考える．また，この時 \(v_x=\frac{x_q}{\sin{\left(\frac{\theta}{2}\right)}}, \;v_y=\frac{y_q}{\sin{\left(\frac{\theta}{2}\right)}}, \;v_z=\frac{z_q}{\sin{\left(\frac{\theta}{2}\right)}}, \;v_x^2+v_y^2+v_z^2=1\)となるような\(v_x, v_y, v_z\)を用意する． $$ \begin{eqnarray} w_q^2+x_q^2+y_q^2+z_q^2 &=& \left( \cos^2{ \left( \frac{\theta}{2} \right) } +\left( \sin{ \left( \frac{\theta}{2} \right) } v_x \right)^2 +\left( \sin{ \left( \frac{\theta}{2} \right) } v_y \right)^2 +\left( \sin{ \left( \frac{\theta}{2} \right) } v_z \right)^2 \right) \\&=&\left( \cos^2{ \left( \frac{\theta}{2} \right) } +\sin^2{ \left( \frac{\theta}{2} \right) } \left( v_x^2+v_y^2+v_z^2 \right) \right) \\&=&\left( \cos^2{ \left( \frac{\theta}{2} \right) } +\sin^2{ \left( \frac{\theta}{2} \right) } \right) \\&=&1 \\w_{qpq^{-1}}&=&w_p \\&=&0 \\x_{qpq^{-1}}&=&(w_q^2+x_q^2-y_q^2-z_q^2)x_p+2(x_qy_q-w_qz_q)y_p+2(w_qy_q+x_qz_q)z_p \\&=& \left( \cos^2{ \left( \frac{\theta}{2} \right) } +\left( \sin{ \left( \frac{\theta}{2} \right) } v_x \right)^2 -\left( \sin{ \left( \frac{\theta}{2} \right) } v_y \right)^2 -\left( \sin{ \left( \frac{\theta}{2} \right) } v_z \right)^2 {\color{red} -2\left( \sin{ \left( \frac{\theta}{2} \right) } v_x \right)^2 +2\left( \sin{ \left( \frac{\theta}{2} \right) } v_x \right)^2 } \right)x_p +2\left( \sin{ \left( \frac{\theta}{2} \right) } v_x \sin{ \left( \frac{\theta}{2} \right) } v_y - \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } v_z \right)y_p +2\left( \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } v_y + \sin{ \left( \frac{\theta}{2} \right) } v_x \sin{ \left( \frac{\theta}{2} \right) } v_z \right)z_p \\&=& \left( \cos^2{ \left( \frac{\theta}{2} \right) } - \sin^2{ \left( \frac{\theta}{2} \right) } \left( v_x^2 +v_y^2 +v_z^2 \right) +2\sin^2{ \left( \frac{\theta}{2} \right) } v_x^2 \right)x_p +2 \sin^2{ \left( \frac{\theta}{2} \right) } v_xv_yy_p -2 \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } v_zy_p +2 \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } v_yz_p +2 \sin^2{ \left( \frac{\theta}{2} \right) } v_xv_zz_p \\&=& \left( \cos^2{ \left( \frac{\theta}{2} \right) } - \sin^2{ \left( \frac{\theta}{2} \right) }\cdot1 \right)x_p +2\sin^2{ \left( \frac{\theta}{2} \right) } v_x^2x_p +2 \sin^2{ \left( \frac{\theta}{2} \right) } \left(v_xv_yy_p + v_xv_zz_p\right) +2 \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } \left(v_yz_p - v_zy_p\right) \\&=& \left( \cos^2{ \left( \frac{\theta}{2} \right) } - \sin^2{ \left( \frac{\theta}{2} \right) } \right)x_p +2 \sin^2{ \left( \frac{\theta}{2} \right) } \left(v_x^2x_p + v_xv_yy_p + v_xv_zz_p\right) +2 \cos{ \left( \frac{\theta}{2} \right) } \sin{ \left( \frac{\theta}{2} \right) } \left(v_yz_p - v_zy_p\right) \\&=& \cos{\left(\theta\right)}x_p +\left(1-\cos{\left(\theta\right)}\right)\left(\left(v_xx_p + v_yy_p + v_zz_p\right)v_x\right) +\sin{\left(\theta\right)}\left(v_yz_p - v_zy_p\right) \\&&\;\cdots\;\cos^2{\left(\frac{\theta}{2}\right)}-\sin^2{\left(\frac{\theta}{2}\right)} =\cos{\left(\frac{\theta}{2}+\frac{\theta}{2}\right)} =\cos{\left(\theta\right)} \\&&\;\cdots\;2\sin^2{\left(\frac{\theta}{2}\right)} =\sin^2{\left(\frac{\theta}{2}\right)}+\sin^2{\left(\frac{\theta}{2}\right)} =\left(1-\cos^2{\left(\frac{\theta}{2}\right)}\right)+\sin^2{\left(\frac{\theta}{2}\right)} =1-\left(\cos^2{\left(\frac{\theta}{2}\right)}-\sin^2{\left(\frac{\theta}{2}\right)}\right) =1-\cos{\left(\theta\right)} \\&&\;\cdots\;2\cos{\left(\frac{\theta}{2}\right)}\sin{\left(\frac{\theta}{2}\right)} =\sin{\left(\frac{\theta}{2}+\frac{\theta}{2}\right)} =\sin{\left(\theta\right)} \\&=& \left(v_xx_p + v_yy_p + v_zz_p\right)v_x +\cos{\left(\theta\right)}\left(x_p - \left(v_xx_p + v_yy_p + v_zz_p\right)v_x\right) +\sin{\left(\theta\right)}\left(v_yz_p - v_zy_p\right) \\&=&\left(\mathbf{V}\cdot \mathbf{V}_p\right)v_x +\cos{\left(\theta\right)}\left(x_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_x\right) +\sin{\left(\theta\right)}\left(v_yz_p - v_zy_p\right) \;\cdots\;\mathbf{V}=(v_x, v_y, v_z) , \mathbf{V}_p=(x_p, y_p, z_p) \\y_{qpq^{-1}}&=&2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p \\&=& \left(v_xx_p + v_yy_p + v_zz_p\right)v_y +\cos{\left(\theta\right)}\left(y_p - \left(v_xx_p + v_yy_p + v_zz_p\right)v_y\right) +\sin{\left(\theta\right)}\left(v_zx_p - v_xz_p\right) \\&=& \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_y +\cos{\left(\theta\right)}\left(y_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_y\right) +\sin{\left(\theta\right)}\left(v_zx_p - v_xz_p\right) \\z_{qpq^{-1}}&=&2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p \\&=& \left(v_xx_p + v_yy_p + v_zz_p\right)v_z +\cos{\left(\theta\right)}\left(z_p - \left(v_xx_p + v_yy_p + v_zz_p\right)v_z\right) +\sin{\left(\theta\right)}\left(v_xy_p - v_yx_p\right) \\&=& \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_z +\cos{\left(\theta\right)}\left(z_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_z\right) +\sin{\left(\theta\right)}\left(v_xy_p - v_yx_p\right) \end{eqnarray} $$ $$ \begin{eqnarray} \mathbf{P}_{qpq^{-1}} &=&w_{qpq^{-1}}+x_{qpq^{-1}}i+y_{qpq^{-1}}j+z_{qpq^{-1}}k \\&=&0 \\&&+\left( \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_x +\cos{\left(\theta\right)}\left(x_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_x\right) +\sin{\left(\theta\right)}\left(v_yz_p - v_zy_p\right) \right)i \\&&+\left( \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_y +\cos{\left(\theta\right)}\left(y_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_y\right) +\sin{\left(\theta\right)}\left(v_zx_p - v_xz_p\right) \right)j \\&&+\left( \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_z +\cos{\left(\theta\right)}\left(z_p - \left(\mathbf{V}\cdot \mathbf{V}_p\right)v_z\right) +\sin{\left(\theta\right)}\left(v_xy_p - v_yx_p\right) \right)k \\&=& \left(\mathbf{V}\cdot \mathbf{V}_p\right)\left(v_xi+v_yj+v_zk\right) +\cos{\left(\theta\right)}\left(x_pi+y_pj+z_pk\right) -\cos{\left(\theta\right)}\left(\mathbf{V}\cdot \mathbf{V}_p\right)\left(v_xi+v_yj+v_zk\right) +\sin{\left(\theta\right)}\left(\left(v_yz_p - v_zy_p\right)i+\left(v_zx_p - v_xz_p\right)j\left(v_xy_p - v_yx_p\right)k\right) \\&=& \left(\mathbf{V}\cdot \mathbf{V}_p\right)\mathbf{V} +\cos{\left(\theta\right)}\mathbf{V}_p -\cos{\left(\theta\right)}\left(\mathbf{V}\cdot \mathbf{V}_p\right)\mathbf{V} +\sin{\left(\theta\right)}\left(\mathbf{V}\times\mathbf{V}_p\right) \\&=& \left(\mathbf{V}\cdot \mathbf{V}_p\right)\mathbf{V} +\cos{\left(\theta\right)}\left( \mathbf{V}_p-\left(\mathbf{V}\cdot \mathbf{V}_p\right) \right)\mathbf{V} +\sin{\left(\theta\right)}\left(\mathbf{V}\times\mathbf{V}_p\right) \end{eqnarray} $$ これはベクトル\(\mathbf{V}_p=(x_p, y_p, z_p)\)を，回転軸\(\mathbf{V}=(v_x, v_y, v_z)\)周りに\(\theta\)だけ回す変換となる．
$$ \begin{eqnarray} \mathbf{q}\mathbf{p}\mathbf{q}^{-1} &\leftrightarrow& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} w_q+x_qi&y_q+z_qi\\ -(y_q-z_qi)&w_q-x_qi\\ \end{bmatrix} \begin{bmatrix} w_p+x_pi&y_p+z_pi\\ -(y_p-z_pi)&w_p-x_pi\\ \end{bmatrix} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} (w_q+x_qi)(w_p+x_pi) +(y_q+z_qi)(-(y_p-z_pi)) & (w_q+x_qi)(y_p+z_pi) +(y_q+z_qi)(w_p-x_pi) \\ (-(y_q-z_qi))(w_p+x_pi) +(w_q-x_qi)(-(y_p-z_pi)) & (-(y_q-z_qi))(y_p+z_pi) +(w_q-x_qi)(w_p-x_pi) \\ \end{bmatrix} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} w_qw_p+w_qx_pi+x_qiw_p+x_qix_pi + y_q(-y_p)+y_qz_pi+z_qi(-y_p)+z_qiz_pi & w_qy_p+w_qz_pi+x_qiy_p+x_qiz_pi + y_qw_p+y_q(-x_pi)+z_qiw_p+z_qi(-x_pi) \\ (-y_q)w_p+(-y_q)x_pi+z_qiw_p+z_qix_pi + w_q(-y_p)+w_qz_pi+(-x_qi)(-y_p)+(-x_qi)z_pi & (-y_q)y_p+(-y_q)z_pi+z_qiy_p+z_qiz_pi + w_qw_p+w_q(-x_pi)+(-x_qi)w_p+(-x_qi)(-x_pi) \\ \end{bmatrix} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i & w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i \\ -(w_qy_p -x_qz_p +y_qw_p +z_qx_p -(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i & w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i \\ \end{bmatrix} \begin{bmatrix} w_q-x_qi&-y_q-z_qi\\ -(-y_q+z_qi)&w_q+x_qi\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q-x_qi) +(w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(-(-y_q+z_qi)) & (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-y_q-z_qi) +(w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(w_q+x_qi)\\ (-(w_qy_p -x_qz_p +y_qw_p +z_qx_p -(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(w_q-x_qi) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-(-y_q+z_qi)) &(-(w_qy_p -x_qz_p +y_qw_p +z_qx_p -(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-y_q-z_qi) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q+x_qi)\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q-x_qi) +(w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(y_q-z_qi)) & (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-y_q-z_qi) +(w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(w_q+x_qi)\\ (-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(w_q-x_qi) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(y_q-z_qi) &(-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-y_q-z_qi) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q+x_qi)\\ \end{bmatrix} \\&=& \frac{1}{w_q^2+x_q^2+y_q^2+z_q^2} \begin{bmatrix} (w_q^2+x_q^2+y_q^2+z_q^2) w_p +( (w_q^2+x_q^2-y_q^2-z_q^2) x_p +2(x_qy_q-w_qz_q) y_p +2(w_qy_q+x_qz_q) z_p )i & 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p +( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i \\ -( 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p -( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i ) & (w_q^2+x_q^2+y_q^2+z_q^2)w_p -( (w_p^2+x_q^2-y_q^2-z_q^2)x_p+2(x_qy_q-w_qz_q)y_p+2(w_qy_q+x_qz_q)z_p )i \\ \end{bmatrix} \\&&\;\cdots\;各要素の式展開は以降に添付 \end{eqnarray} $$
(1,1)要素
(1,1)要素の第一項$$ \begin{eqnarray} (w_qw_p -x_qx_p -y_qy_p -z_qz_p + (w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q-x_qi) &=& (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q) + (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-x_qi) \\&=& w_pw_qw_q -w_qx_px_q -w_qy_py_q -w_qz_pz_q + w_qx_px_q +w_px_qx_q +x_qy_qz_p -x_qy_pz_q +( w_qw_qx_p +w_pw_qx_q +w_qy_qz_p -w_qy_pz_q -w_pw_qx_q +x_px_qx_q +x_qy_py_q +x_qz_pz_q )i \end{eqnarray} $$ (1,1)要素の第二項
$$ \begin{eqnarray} (w_qy_p +y_qw_p +z_qx_p -x_qz_p + (w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(-(-y_q+z_qi)) &=& (w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(y_q) + (w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(-z_qi) \\&=& w_qy_py_q +w_py_qy_q +x_py_qz_q -x_qy_qz_p + w_qz_pz_q +w_pz_qz_q +x_qy_pz_q -x_py_qz_q +( w_qy_qz_p +w_py_qz_q +x_qy_py_q -x_py_qy_q -w_qy_pz_q -w_py_qz_q -x_pz_qz_q +x_qz_pz_q )i \end{eqnarray} $$ (1,1)要素の第一，二項の和
$$ \begin{eqnarray} (w_qw_p -x_qx_p -y_qy_p -z_qz_p + (w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q-x_qi)\\ +(w_qy_p +y_qw_p +z_qx_p -x_qz_p + (w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(-(-y_q+z_qi)) &=& w_pw_qw_q -w_qx_px_q -w_qy_py_q -w_qz_pz_q + w_qx_px_q +w_px_qx_q +x_qy_qz_p -x_qy_pz_q + w_qy_py_q +w_py_qy_q +x_py_qz_q -x_qy_qz_p + w_qz_pz_q +w_pz_qz_q +x_qy_pz_q -x_py_qz_q +( w_qw_qx_p +w_pw_qx_q +w_qy_qz_p -w_qy_pz_q -w_pw_qx_q +x_px_qx_q +x_qy_py_q +x_qz_pz_q +w_qy_qz_p +w_py_qz_q +x_qy_py_q -x_py_qy_q -w_qy_pz_q -w_py_qz_q -x_pz_qz_q +x_qz_pz_q )i \\ &=& (w_q^2+x_q^2+y_q^2+z_q^2) w_p +( (w_q^2+x_q^2-y_q^2-z_q^2) x_p +2(x_qy_q-w_qz_q) y_p +2(w_qy_q+x_qz_q) z_p )i \end{eqnarray} $$
(1,2)要素
(1,2)要素の第一項$$ \begin{eqnarray} (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-y_q-z_qi) &=& (w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-y_q) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p +(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(-z_qi) \\&=& -w_pw_qy_q +x_px_qy_q +y_py_qy_q +z_pz_qy_q +w_qx_pz_q +w_px_qz_q +y_qz_pz_q -y_pz_qz_q +(-w_qx_py_q -w_px_qy_q -y_qy_qz_p +y_py_qz_q -w_pw_qz_q +x_px_qz_q +y_py_qz_q +z_pz_qz_q)i \end{eqnarray} $$ (1,2)要素の第二項
$$ \begin{eqnarray} ( w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(w_q+x_qi) &=& ( w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)w_q +( w_qy_p +y_qw_p +z_qx_p -x_qz_p +(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)x_qi \\&=& w_qw_qy_p +w_pw_qy_q +w_qx_pz_q -w_qx_qz_p -w_qx_qz_p -w_px_qz_q -x_qx_qy_p +x_px_qy_q +( w_qw_qz_p +w_pw_qz_q +w_qx_qy_p -w_qx_py_q +w_qx_qy_p +w_px_qy_q +x_px_qz_q -x_qx_qz_p )i \end{eqnarray} $$ (1,2)要素の第一，二項の和
$$ \begin{eqnarray} (w_qw_p-x_qx_p -y_qy_p -z_qz_p+(w_qx_p+x_qw_p +y_qz_p -z_qy_p)i)(-y_q-z_qi)\\ + (w_qy_p +y_qw_p +z_qx_p -x_qz_p+(w_qz_p +z_qw_p +x_qy_p -y_qx_p)i)(w_q+x_qi) &=& -w_pw_qy_q +x_px_qy_q +y_py_qy_q +z_pz_qy_q +w_qx_pz_q +w_px_qz_q +y_qz_pz_q -y_pz_qz_q +w_qw_qy_p +w_pw_qy_q +w_qx_pz_q -w_qx_qz_p -w_qx_qz_p -w_px_qz_q -x_qx_qy_p +x_px_qy_q +( -w_qx_py_q -w_px_qy_q -y_qy_qz_p +y_py_qz_q -w_pw_qz_q +x_px_qz_q +y_py_qz_q +z_pz_qz_q +w_qw_qz_p +w_pw_qz_q +w_qx_qy_p -w_qx_py_q +w_qx_qy_p +w_px_qy_q +x_px_qz_q -x_qx_qz_p )i \\&=& 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p +( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i \end{eqnarray} $$
(2,1)要素
(2,1)要素の第一項$$ \begin{eqnarray} (-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(w_q-x_qi) &=& -w_qy_p +x_qz_p -y_qw_p -z_qx_p +( w_qz_p +x_qy_p -y_qx_p +z_qw_p )i )(w_q) -w_qy_p +x_qz_p -y_qw_p -z_qx_p +( w_qz_p +x_qy_p -y_qx_p +z_qw_p )i )(-x_qi) \\&=& -w_qy_pw_q +x_qz_pw_q -y_qw_pw_q -z_qx_pw_q +w_qz_px_q +x_qy_px_q -y_qx_px_q +z_qw_px_q +( w_qz_pw_q +x_qy_pw_q -y_qx_pw_q +z_qw_pw_q +w_qy_px_q -x_qz_px_q +y_qw_px_q +z_qx_px_q )i \end{eqnarray} $$ (2,1)要素の第二項
$$ \begin{eqnarray} ( w_qw_p -x_qx_p -y_qy_p -z_qz_p -( w_qx_p +x_qw_p +y_qz_p -z_qy_p )i )(y_q-z_qi) &=& ( w_qw_p -x_qx_p -y_qy_p -z_qz_p -( w_qx_p +x_qw_p +y_qz_p -z_qy_p )i )(y_q) +( w_qw_p -x_qx_p -y_qy_p -z_qz_p -( w_qx_p +x_qw_p +y_qz_p -z_qy_p )i )(-z_qi) \\&=& w_qw_py_q -x_qx_py_q -y_qy_py_q -z_qz_py_q -w_qx_pz_q -x_qw_pz_q -y_qz_pz_q +z_qy_pz_q +( -w_qx_py_q -x_qw_py_q -y_qz_py_q +z_qy_py_q -w_qw_pz_q +x_qx_pz_q +y_qy_pz_q +z_qz_pz_q )i \end{eqnarray} $$ (2,1)要素の第一，二項の和
$$ \begin{eqnarray} ( -w_qy_p +x_qz_p -y_qw_p -z_qx_p +( w_qz_p +x_qy_p -y_qx_p +z_qw_p )i )(w_q-x_qi) \\ +( w_qw_p -x_qx_p -y_qy_p -z_qz_p -( w_qx_p +x_qw_p +y_qz_p -z_qy_p )i )(y_q-z_qi) &=& -w_qy_pw_q +x_qz_pw_q -y_qw_pw_q -z_qx_pw_q +w_qz_px_q +x_qy_px_q -y_qx_px_q +z_qw_px_q +( w_qz_pw_q +x_qy_pw_q -y_qx_pw_q +z_qw_pw_q +w_qy_px_q -x_qz_px_q +y_qw_px_q +z_qx_px_q )i + w_qw_py_q -x_qx_py_q -y_qy_py_q -z_qz_py_q -w_qx_pz_q -x_qw_pz_q -y_qz_pz_q +z_qy_pz_q +( -w_qx_py_q -x_qw_py_q -y_qz_py_q +z_qy_py_q -w_qw_pz_q +x_qx_pz_q +y_qy_pz_q +z_qz_pz_q )i \\&=& -w_qy_pw_q +x_qz_pw_q -y_qw_pw_q -z_qx_pw_q +w_qz_px_q +x_qy_px_q -y_qx_px_q +z_qw_px_q +w_qw_py_q -x_qx_py_q -y_qy_py_q -z_qz_py_q -w_qx_pz_q -x_qw_pz_q -y_qz_pz_q +z_qy_pz_q +( w_qz_pw_q +x_qy_pw_q -y_qx_pw_q +z_qw_pw_q +w_qy_px_q -x_qz_px_q +y_qw_px_q +z_qx_px_q -w_qx_py_q -x_qw_py_q -y_qz_py_q +z_qy_py_q -w_qw_pz_q +x_qx_pz_q +y_qy_pz_q +z_qz_pz_q )i \\&=& -2(w_qz_q+x_qy_q)x_p(-w_q^2+x_q^2-y_q^2+z_q^2)y_p+2(w_qx_q-y_qz_q)z_p +( (w_q^2-x_q^2-y_q^2+z_q^2)z_p+2(w_qx_q+y_qz_q)y_p+2(x_qz_q-w_qy_q)x_p )i \\&=& -( 2(w_qz_q+x_qy_q)x_p+(w_q^2-x_q^2+y_q^2-z_q^2)y_p+2(y_qz_q-w_qx_q)z_p -( 2(x_qz_q-w_qy_q)x_p+2(w_qx_q+y_qz_q)y_p+(w_q^2-x_q^2-y_q^2+z_q^2)z_p )i ) \end{eqnarray} $$
(2,2)要素
(2,2)要素の第一項$$ \begin{eqnarray} (-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-y_q-z_qi) &=& (-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-y_q) +(-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-z_qi) \\&=& w_qy_py_q -x_qz_py_q +y_qw_py_q +z_qx_py_q +w_qz_pz_q +x_qy_pz_q -y_qx_pz_q +z_qw_pz_q +( -w_qz_py_q -x_qy_py_q +y_qx_py_q -z_qw_py_q +w_qy_pz_q -x_qz_pz_q +y_qw_pz_q +z_qx_pz_q )i \end{eqnarray} $$ (2,2)要素の第二項
$$ \begin{eqnarray} (w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q+x_qi) &=& (w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q) +(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(x_qi) \\&=& w_qw_pw_q -x_qx_pw_q -y_qy_pw_q -z_qz_pw_q +w_qx_px_q +x_qw_px_q +y_qz_px_q -z_qy_px_q +( -w_qx_pw_q -x_qw_pw_q -y_qz_pw_q +z_qy_pw_q +w_qw_px_q -x_qx_px_q -y_qy_px_q -z_qz_px_q )i \end{eqnarray} $$ (2,1)要素の第一，二項の和
$$ \begin{eqnarray} (-w_qy_p +x_qz_p -y_qw_p -z_qx_p +(w_qz_p +x_qy_p -y_qx_p +z_qw_p)i)(-y_q-z_qi) \\+(w_qw_p -x_qx_p -y_qy_p -z_qz_p -(w_qx_p +x_qw_p +y_qz_p -z_qy_p)i)(w_q+x_qi) &=& (w_qy_py_q -x_qz_py_q +y_qw_py_q +z_qx_py_q +w_qz_pz_q +x_qy_pz_q -y_qx_pz_q +z_qw_pz_q +( -w_qz_py_q -x_qy_py_q +y_qx_py_q -z_qw_py_q +w_qy_pz_q -x_qz_pz_q +y_qw_pz_q +z_qx_pz_q )i) + (w_qw_pw_q -x_qx_pw_q -y_qy_pw_q -z_qz_pw_q +w_qx_px_q +x_qw_px_q +y_qz_px_q -z_qy_px_q +( -w_qx_pw_q -x_qw_pw_q -y_qz_pw_q +z_qy_pw_q +w_qw_px_q -x_qx_px_q -y_qy_px_q -z_qz_px_q )i) \\&=& w_qy_py_q -x_qz_py_q +y_qw_py_q +z_qx_py_q +w_qz_pz_q +x_qy_pz_q -y_qx_pz_q +z_qw_pz_q +w_qw_pw_q -x_qx_pw_q -y_qy_pw_q -z_qz_pw_q +w_qx_px_q +x_qw_px_q +y_qz_px_q -z_qy_px_q +( -w_qz_py_q -x_qy_py_q +y_qx_py_q -z_qw_py_q +w_qy_pz_q -x_qz_pz_q +y_qw_pz_q +z_qx_pz_q -w_qx_pw_q -x_qw_pw_q -y_qz_pw_q +z_qy_pw_q +w_qw_px_q -x_qx_px_q -y_qy_px_q -z_qz_px_q )i) \\&=& (w_q^2+x_q^2+y_q^2+z_q^2)w_p +( (-w_p^2-x_q^2+y_q^2+z_q^2)x_p+2(w_qz_q-x_qy_q)y_p-2(w_qy_q+x_qz_q)z_p )i \\&=& (w_q^2+x_q^2+y_q^2+z_q^2)w_p -( (w_p^2+x_q^2-y_q^2-z_q^2)x_p+2(x_qy_q-w_qz_q)y_p+2(w_qy_q+x_qz_q)z_p )i \end{eqnarray} $$
