How to write this term compactly?

Question

How can I write this term in a compact form where $a$ only appears once on the RHS (in particular without cases)?

$T(a) = \begin{cases} a^2 &,\text{ if $a \leq 0$}\\ 2a^2 &,\text{ if $a > 0$}\\ \end{cases}$

I have already thought about $T(a) = \max\{\sqrt{2}a,|a|\}^2$ or $T(a) = \frac{3+\text{sgn}(a)}{2}a^2$, but in both cases $a$ appears twice.

Answer 1

After discussing the problem with a brilliant friend we came up with the following solution:

$T(a) = \left[\Re\left((\sqrt[4]{2}-i)\sqrt{a}\right)\right]^4$

However, I am still up for further suggestions!

Answer 2

Another class of solutions to this problem is the "trivial substitution class".

Examples

$$T(a) = 2\int_0^a x\cdot(\max\{x,0\}+1)dx$$

or even more trivial

$$T(a) = \max\{\sqrt{2}x,|x|\}^2\bigg|_{x=a}$$