To highlight asymmetries in data distributions for pairwise causal discovery, Mitrovic et al. introduced kernel embeddings based on the conditional distributions $P_{Y | X}$ and $P_{X | Y}$ rather than the joint distribution. The proposed conditional embedding utilizes a Gaussian kernel and an $\alpha$ quantity for conditioning: $\mu_{k,M}(P_{S_j}) = \{ \sum_{j=1}^{n_j} \alpha_j (y)k_m(\cdot , x_j), \sum_{j=1}^{n_j} \alpha_j (x) k_m(\cdot , y_j) \}_{m\in M}$. Here, $\alpha (y) = (L + n\lambda I)^{-1} l_y$, with $L = [l(y_i, y_j)]_{i,j=1}^n$ and $l_y = [l(y_1, y), \dots, l(y_n, y)]^T$. The term $\alpha (\cdot) = [\alpha_1 (\cdot), \dots, \alpha_n (\cdot)]^T$, $\lambda$ is a regularization parameter, $I$ is the identity matrix, and $M$ is the set of parameters for the kernel $k$.