It may be required for relativistic quantum mechanics, so this should be kept in mind. However, all of the components of this equation are quaternions, not separate structures such as an operator belonging to a group and a vector. This might make a subtle but significant difference for the mathematical structure of the theory, a point that will not be investigated here. If B hat is an observable, then this operator can be defined in terms of the scalar variable a. Operators A and B are linear. It is hard to shuffle quaternions or their operators around.
Real scalars commute with any quaternion and are their own conjugates.
Operators that generate such scalars can move around. Look at ways to express the expectation value of A. Define a new operator A' based on A whose expectation value is always zero. Define the square of the operator in a way designed to link up with the standard deviation. In the section on bracket notation, the Schwarz inequality for quaternions was shown. The Schwarz inequality applies to quaternions, not quaternion operators.