Group of unit quaternions

Definition
This group is denoted $$S^3, SU(2), S^0(\mathbb{H}), S^1(\mathbb{C})$$ and is defined in a number of equivalent ways.

As the group of unit quaternions
Denote by $$\mathbb{H}$$ the division ring of Hamiltonian quaternions. The group we are interested in is the multiplicative subgroup of $$\mathbb{H}^*$$ comprising those unit quaternions $$a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$$ satisfying $$a^2 + b^2 + c^2 + d^2 = 1$$. Note that $$a,b,c,d \in \R$$ (and are allowed to be equal). Explicitly, the multiplication is given by:

$$(a_1 + b_1\mathbf{i} + c_1\mathbf{j} + d_1\mathbf{k})(a_2 + b_2\mathbf{i} + c_2\mathbf{j} + d_2\mathbf{k}) = (a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2) + (a_1b_2 + a_2b_1 + c_1d_2 - c_2d_1)\mathbf{i} + (a_1c_2 + a_2c_1 + d_1b_2 - d_2b_1)\mathbf{j} + (a_1d_2 + a_2d_1 + b_1c_2 - b_2c_1)\mathbf{k}$$

The identity element is:

$$1 + 0\mathbf{i} + 0\mathbf{j} + 0\mathbf{k}$$

The inverse is given by:

$$(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k})^{-1} = a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k}$$

As the special unitary group
The group can also be defined as the special unitary group of degree two over the field of complex numbers. It is denoted $$SU(2)$$ or $$SU(2,mathbb{C})$$.

Structures
The group has the following structures:


 * It is a real Lie group (note that it is not a complex Lie group).
 * It is a linear algebraic group over the field of real numbers (note that it is not algebraic over the complex numbers).
 * It is a topological group.

Topological/Lie group properties
The topology here is the subspace topology from the Euclidean topology on the set of all matrices, which is identified with the Euclidean space $$\R^4$$.