Double-Cross Product between 2 Dyads/Dyadics

1. For calculating Double-Cross Product between 2 Dyads/Dyadics the Order of the both Dyads/Dyadics must be 3. The result of Double-Cross Product between 2 Dyads/Dyadics is a Dyad/Dyadic.
2. Given a Dyad \(\overleftrightarrow{AB}\) formed of Vectors \(\vec{A}\) and \(\vec{B}\) and another Dyad \(\overleftrightarrow{CD}\) formed of Vectors \(\vec{C}\) and \(\vec{D}\), the following relations holds true
   
   \(\overleftrightarrow{AB} {}_\times^\times \overleftrightarrow{CD}= -(\vec{A} \times \overleftrightarrow{CD}) \times \vec{B}= (\vec{A} \times \vec{C})\otimes_{T}(\vec{B} \times \vec{D})\)
   
   \(\overleftrightarrow{CD} {}_\times^\times \overleftrightarrow{AB}= -(\vec{C} \times \overleftrightarrow{AB}) \times \vec{D}= (\vec{C} \times \vec{A})\otimes_{T}(\vec{D} \times \vec{B})\)
3. Given a Dyadic Polynomial Matrix \(\overleftrightarrow{AB}\) formed by addition of Dyads \(\overleftrightarrow{A_1B_1}\), \(\overleftrightarrow{A_2B_2}\), ... , \(\overleftrightarrow{A_KB_K}\) and another Dyadic Polynomial Matrix \(\overleftrightarrow{CD}\) formed by addition of Dyads \(\overleftrightarrow{C_1D_1}\), \(\overleftrightarrow{C_2D_2}\), ... , \(\overleftrightarrow{C_ND_N}\), the following relations hold true
   
   \(\overleftrightarrow{AB} {}_\times^\times \overleftrightarrow{CD}= -\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \times \overleftrightarrow{C_jD_j}) \times \vec{B_i}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \times \vec{C_j})\otimes_{T}(\vec{B_i} \times \vec{D_j})\)
   
   \(\overleftrightarrow{CD} {}_\times^\times \overleftrightarrow{AB}= -\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{C_j} \times \overleftrightarrow{A_iB_i}) \times \vec{D_j}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{C_j} \times \vec{A_i})\otimes_{T}(\vec{D_j} \times \vec{B_i})\)
4. The Double-Cross Product between 2 Dyads/Dyadics is Commutative, that is for any 2 Dyad/Dyadic \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)
   
   \(\overleftrightarrow{AB} {}_\times^\times \overleftrightarrow{CD} = \overleftrightarrow{CD} {}_\times^\times \overleftrightarrow{AB}\)
5. Given a Dyad \(\overleftrightarrow{AB}\) formed from Vectors \(\vec{A}\) and \(\vec{B}\) and Identity Dyadic \(\overleftrightarrow{I}\), following relation holds true
   
   \(\overleftrightarrow{I} {}_\times^\times \overleftrightarrow{AB}=\overleftrightarrow{AB} {}_\times^\times \overleftrightarrow{I}=(\overleftrightarrow{AB} :\overleftrightarrow{I})\overleftrightarrow{I} - \overleftrightarrow{AB}^T\)