Nondegenerate does not mean definite

Consider the bilinear form $B:{\mathbb{R}}^4\times {\mathbb{R}}^4\to \mathbb{R}$ defined by $B(x,y)=x_1{y}_3+x_2{y}_4-x_3{y}_1-x_4{y}_2$, where $x=(x_1,x_2,x_3,x_4)$ and $y=(y_1,y_2,y_3,y_4)$ in ${\mathbb{R}}^4$. Let A denote the matrix of B with respect to the standard ordered basis of ${\mathbb{R}}^4$. Which of the following statements is true?