If I load and then unload NCAlgebra package, my code snippet will not work properly.

[hongyi-zhao]

My testing is as follows:

1. First, test without loading NCAlgebra package, my code snippet works smoothly:

In[1]:= (*
The following method doesn't rely on NCAlgebra package.

Unable to Simplify non-commutative Expression Further in Mathematica.
https://community.wolfram.com/groups/-/m/t/3076922*)
ClearAll["Global`*"]
Unprotect[NonCommutativeMultiply];
a_ ** (b_?NumberQ c_) := b*(a ** c)
(b_?NumberQ c_) ** a_ := b*(c ** a)
a_ ** (b_ + c_) := a ** b + a ** c
(b_ + c_) ** a_ := b ** a + c ** a
comm[x_, y_] := x ** y - y ** x
anticomm[x_, y_] := x ** y + y ** x
A ** anticomm[C, B] ** D - A ** C ** anticomm[D, B] + 
  anticomm[C, A] ** D ** B - C ** anticomm[D, A] ** B // Simplify
% == comm[A ** B, C ** D] // FullSimplify

Out[9]= A ** B ** C ** D - C ** D ** A ** B

Out[10]= True

2. Then, I try to load NCAlgebra package, and then unload it by ClearAll as follows:

<< NC`
<< NCAlgebra`

NC::Directory: You are using a paclet version of NCAlgebra.

NCAlgebra::SmallCapSymbolsNonCommutative: All lower cap single letter symbols (e.g. a,b,c,...) were set as noncommutative.

In[15]:= (*
The following method doesn't rely on NCAlgebra package.

Unable to Simplify non-commutative Expression Further in Mathematica.
https://community.wolfram.com/groups/-/m/t/3076922*)
ClearAll["Global`*"]
Unprotect[NonCommutativeMultiply];
a_ ** (b_?NumberQ c_) := b*(a ** c)
(b_?NumberQ c_) ** a_ := b*(c ** a)
a_ ** (b_ + c_) := a ** b + a ** c
(b_ + c_) ** a_ := b ** a + c ** a
comm[x_, y_] := x ** y - y ** x
anticomm[x_, y_] := x ** y + y ** x
A ** anticomm[C, B] ** D - A ** C ** anticomm[D, B] + 
  anticomm[C, A] ** D ** B - C ** anticomm[D, A] ** B // Simplify
% == comm[A ** B, C ** D] // FullSimplify

During evaluation of In[15]:= ClearAll::wrsym: Symbol T is Protected.

Out[23]= 0

Out[24]= True

As you can see, now the result is 0. Any tips for fixing this problem will be appreciated.

See here for the related discussion.

Regards,
Zhao

[mcdeoliveira]

You assume that ClearAll "unloads" NCAlgebra but this is far from it.