How to prove vector identities WITHOUT using levi civita ?

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The discussion focuses on deriving vector identities without using the Levi-Civita symbol or multiplying components directly. The user successfully derived identities such as (AXB).(CXD) and is seeking methods to improve their ability to quickly recognize and prove other vector identities, particularly in preparation for an upcoming exam. They mention specific identities they aim to prove, including ∇. (AXB) and ∇x(AxB). The user also shares links to lectures that may assist in understanding these concepts better. Ultimately, they express a desire to leverage their strengths while preparing for the test.
darksilence
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Mentor note: Thread moved from homework sections as being a better fit in the math technical section.
Multiplying components of both sides are also off limits.
I am trying to derive vector identities on introduction chapters in various EMT books. For example : (AXB).(CXD) = (A.C)(B.D) - (A.D)(B.C)
After a few hours i noticed B.(CXD) = C.(DXB) and replaced B's with AXB's its Done.
AX(BX(CXD)) was even simpler didnt take any time at all.
I want to do that to
∇. (AXB) = B.(∇xA) - A.(∇xB)
∇x(AxB) = ...
∇(A.B) = ...
∇x(∇xA) = ∇(∇.A) - ∇2A etc

So far last 2 days after solving the first two of them just looking them and hoping to see it. What i should do to improve my ability to see them fast ? (I also have to finish half the book in 2-3 weeks before exam so i am hoping to solve this problem in a few days at most.)
 
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Thank you. It wasnt what i wanted at all but some way i didnt imagine they were still helpfull. Instead of working on my weakness i will go on with my strengths.
 

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