The reason for the 2 in the definition of the rotated quadratic cone

MOSEK employs the definition
$$\begin{equation} K^1 := \{ x \mid 2 x_1 x_2 \geq ||x_{3:n}||^2,  x_1, x_2 \geq 0 \} \end{equation}$$
for the rotated quadratic cone, Occasionally users of MOSEK ask why there is the 2 in front of the product $x_1 x_2$.  Why not use the definition
$$\begin{equation} K^2 := \{ x \mid  x_1 x_2 \geq ||x_{3:n}||^2,  x_1, x_2 \geq 0 \} ? \end{equation}$$

The reason is that the dual cone plays an important role and the dual cone of $K^1$ is $K^1$ i.e. it is self-dual. That is pretty! Now the dual cone of $K^2$ is
$$\begin{equation} \{ x \mid 4 s_1 s_2 \geq ||s_{3:n}||^2,  s_1, s_2 \geq 0 \}. \end{equation}$$

Hence, $K^2$ is not self-dual! That is somewhat ugly and inconvenient. 

To summarize the definition $K^1$ for the rotated quadratic cone is preferred because the alternative definition $K^2$ is not self-dual

A couple of historical notes are:

• In the classical paper  by Lobo et. al.  the cone $K^2$ is called a hyperbolic constraint in Section 2.3.  
• MOSEK is highly inspired by the important work of the late Jos Sturm on the code SeDuMi . Now SeDuMi is short for self-dual minimization and for that reason Sturm employs the definition $K^1$ too.