I am reading the book: Infinite-Dimensional Lie Algebras (Kac, third edition) and the article: Affine Lie algebras and the Virasoro algebras I (Wakimoto, link). The formulas they wrote for the Lie bracket $[,]$, normalized standard invariant form $(|)$ of twisted affine Lie algebras of type $X_N^{(r)}$ are contradicted to each other:

**Contradiction1:** In the book, page 139, the bracket is given by

but in the article, page 381, it is given by

Here $X(j)$ means $t^j \otimes X$ and $c_s=rK/m$ (see the article to verify it). They are totally different.

**Contradiction2:** In the book, page 139, if the normalized standard invariant form is defined by

then it contradicts to the Lie bracket in the same page,

since $(d'| [t^i \otimes x, t^j \otimes y]) \ne ([d',t^i \otimes x]| t^j \otimes y)$

So, If are there anyone knows the right formulas for the Lie bracket and normalized standard invariant form for twisted affine Lie algebras mentioned in the Theorem 8.7 in the book of Kac?