Find the critical points of $x^2+y^2+z^2+2xyz$

Find the critical points of the function and specify the nature ofthese points. $$f(x,y,z)=x^2+y^2+z^2+2xyz$$

I already found the critical points $A(0,0,0), B(1,1,-1), C(1,-1,1), D(-1,1,1), E(-1,-1,-1)$. I also proved $A(0,0,0)$ is a saddle point of $f$. I'm stuck at such points $B(1,1,-1)$. The Hessian matrix at $B(1,1,-1)$ is $H=\begin{pmatrix} 2 &-2& 2\\ -2& 2& 2\\ 2& 2 &2\end{pmatrix}$ and $det(\triangle_2)=\begin{vmatrix} 2& -2\\ -2 & 2\end{vmatrix}\ =0.$ So, the matrix is neither positive nor alternating.
How can I determine the nature of $B(1,1-1)$?