  In this article, we give the most genaral form of the quaternions algebra
depending on 3-parameters. We define 3-parameter generalized quaternions
(3PGQs) and study on various properties and applications. Firstly we present
the definiton, the multiplication table another properties of 3PGQs such as
addition-substraction, multiplication and multiplication by scalar operations,
unit and inverse elements, conjugate and norm. We give matrix representation
and Hamilton operators for 3PGQs. We get polar represenation, De Moivre's and
Euler's formulas with the matrix representations for 3PGQs. Besides, we give
relations among the powers of the matrices associated with 3PGQs. Finally, Lie
group and Lie algebra are studied and their matrix representations are shown.
Also the Lie multiplication and the killing bilinear form are given.
