Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that $$ (\alpha + p\delta)^{\vee} = \alpha^{\vee} + \frac{6p}{(\alpha, \alpha)}K, $$ where $K$ is the central element. Are there some references about this formula? I tried to look for this formula in Kac's book. But I am not able to find it. Thank you very much.

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that $$ (\alpha + p\delta)^{\vee} = \alpha^{\vee} + \frac{2p}{(\alpha, \alpha)}K, $$ where $K$ is the central element. Are there some references about this formula? I tried to look for this formula in Kac's book. But I am not able to find it. Thank you very much.

Edit: this formula is given on line 14, page 5 of the paper.