Does $CABA-ABAC=A$ imply $CB=BC$ or $AB=BA$?

Question

Let $A\in GL(2n, \Bbb R)$ be an invertible skew-symmetric matrix such that $A^2+I=0$ and $B,C\in GL(2n, \Bbb R)$ symmetric matrices which $C$ is invertible and $CA=-AC$ and $C^2=\alpha I$ for some $\alpha \in \Bbb R$. Suppose that $A,B$ and $C$ satisfies the following relation: $$CABA-ABAC=A,\tag{1}\label{A}$$

Then does \eqref{A} imply $CB=BC$ or $AB=BA$?

Answer 1

No. Counterexample: $$ A=\pmatrix{0&1\\ -1&0},\ B=\pmatrix{1&-1/2\\ -1/2&-1},\ C=\pmatrix{-1\\ &1}. $$