The following table contains statistics about identities which exist between socalled standard terms of the classical vector analysis in R^{3}. Standard terms formed from k vectors and 1 test vector x are for instance
k + 1 = 4  k + 1 = 5  
<a,b><c,x>
<a,c><b,x> <a,x><b,c> 

Examples of identities between standard terms are for instance
<a,d> b × c + <b,d> c × a + <c,d> a × b = <a,b,c> d (an identity which does not contain a test vector x)
and

=  0 . 
More details can be found on the page of PERMS notebooks where I present the text of a talk (given at SLC46, Lyon, 2001) and the Mathematica notebooks of the calculation of the following table.
k + 1  l  blocks  # ideals  dim  terms  # ident  # summands 
4  (4)
(2,2) 
(4)
(2,2) 
1
1 
1
2 
3  0   
5  (3,1,1)  (3,1,1)  1  6  10  4  4 
6  (6)
(4,2) (2,2,2) 
(6)
(4,2) (2,2,2) 
1
1 1 
1
9 5 
15  0   
7  (5,1,1)
(3,3,1) 
(5,1,1)
(3,3,1) 
1
1 
15
21 
105  69  4, 10, 12, 14, 16, 18 
8  (8)
(6,2) (4,4) (4,2,2) 
(8)
(6,2) (4,4) (4,2,2) 
1
1 1 1 
1
20 14 56 
105  14  24, 36, 40, 44, 50, 52 
9  (7,1,1)
(5,3,1) (3,3,3) 
(7,1,1)
(5,3,1) (3,3,3) 
1
1 1 
28
162 42 
1260  1028  ??? 
k  number of vectors which are used to form standard expressions (k vectors + 1 test vector x) 
l  partitions which are grouping partitions for a given k 
blocks  partitions which denote minimal twosided ideals of the group ring in which the characterizing spaces W have nontrivial projections 
# ideals  number of minimal right ideals which form the projection of W in the twosided ideal denoted by 'blocks' 
dim  dimension of the minimal right ideals from the column '# ideals' 
terms  number of standard terms for a given k 
# ident  number of linearly independent identities for a given k 
# summands  number of summands which I found in a set of linearly independent identities 
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B. Fiedler, 04.09.2022 