Finding a kernel with feature transformation Suppose we have feature transformation $\Phi(x) = [1, x_1, x_2, x_1x_2]$. Now we want to find the kernel corresponding to $\Phi$.

What I have done is using kernel decomposition, we have:
$$
K(x, y) = \Phi(x) .\Phi(y)\\
K(x, y) = 1.1 + x_1y_1 + x_2y_2 + x_1x_2y_1y_2 \\
K(x, y) = 1 + \sum_{i=1}^{N}x_iy_i + \prod_{i=1}^{N}x_iy_i
$$

$x, y$ are in $\Bbb{R}^2$

Is that kernel valid?
Yes, it is valid.

Kernel is just an inner product in the feature space. In fact, here you have explicitly started from the feature transformation and we understand the feature transformation.