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Geometrical interpretation of the left contraction's explicit definition (3.10) in GA for CS

Hi,

The explicit definition of the left contraction is given by the properties (3.7)-(3.11) in “Geometric Algebra for Computer Science”.

Is there a geometrical interpretation for the defining property (3.10)?

Best,
Hongying

P.S.
(3.7): \alpha \rfloor \mathbf{B} = \alpha \mathbf{B}
(3.8): \mathbf{B} \rfloor \alpha = 0 if grad(\mathbf{B})>0
(3.9): \mathbf{a} \rfloor \mathbf{b} = \mathbf{a} \cdot \mathbf{b}
(3.10): \mathbf{a} \rfloor (\mathbf{B} \wedge \mathbf{C}) = (\mathbf{a} \rfloor \mathbf{B}) \wedge \mathbf{C} + (-1)^{\text{grad}(\mathbf{B})} \mathbf{B} \wedge (\mathbf{a} \rfloor{\mathbf{C}})
(3.11): (\mathbf{A} \wedge \mathbf{B}) \rfloor \mathbf{C} = \mathbf{A} \rfloor (\mathbf{B} \rfloor \mathbf{C})