3 edited body

I couldn't write this as a comment and I might be off base, but in Berger's paper Theorem 6 he proves that the list of real Lie groups acting transitively on the quadric $$x_1^2 + \dots + x_{n-h}^2 - x^2_{n-h+1} - \dots - x^2_{n} = 1$$ induced from a linear action on $\mathbb{R}^n$ are modulo a finite number of exceptions
1) $SO(n-h,h)$ $n \geq 2$

2) $T^1 \times SU(m/2-h/2,h/2)$, $SU(m/2-h/2,h/2)$ for some $m$ $m/2 \geq 2$

3) $SP(1) \times SP(m/4-h/2,h/2)$, $T^1 \times SP(m/4-h/2,h/2)$, $SP(m/4-h/2,h/2)$ $m/4 \geq 2$

Edit: since I still think the answer to your question is in Berger's list, I transcribe it completely as it can be found in [2]

signature (p,q): $SO(p,q)$
signature (2p, 2q) $U(p,q)$ and $SU(p,q)$
signature (4p,4q) $SP(1)SP(p,q)$ $SP(p,q)$
signature (n,n) $SO(n,\mathbb{C})$
signature (2n,2n) $SO(n, \mathbb{H})$

$G_2^\mathbb{C}$ $G_2$ and $G_2^2$ signatures (7,7), (7,0) and (4,3)

$Spin(7,\mathbb{C})$ $Spin(7)$ and Spin(4,3) signatures (8,8) (8,0) and (4,4).

you of course are interested in signature (p,1)p,2)

The point is that irreducible holonomies are the ones that act transitively on the unit sphere in the tangent space. And this seems to me as the problem you are asking for.

[2] Proc of Symp in pure mathematics 54 (1993) part 2. On the Holonomy of Lorentzian manifolds. Bergery and Ikemakhen

I couldn't write this as a comment and I might be off base, but in Berger's paper Theorem 6 he proves that the list of real Lie groups acting transitively on the quadric $$x_1^2 + \dots + x_{n-h}^2 - x^2_{n-h+1} - \dots - x^2_{n} = 1$$ induced from a linear action on $\mathbb{R}^n$ are modulo a finite number of exceptions
1) $SO(n-h,h)$ $n \geq 2$

2) $T^1 \times SU(m/2-h/2,h/2)$, $SU(m/2-h/2,h/2)$ for some $m$ $m/2 \geq 2$

3) $SP(1) \times SP(m/4-h/2,h/2)$, $T^1 \times SP(m/4-h/2,h/2)$, $SP(m/4-h/2,h/2)$ $m/4 \geq 2$

Edit: since I still think the answer to your question is in Berger's list, I transcribe it completely as it can be found in [2]

signature (p,q): $SO(p,q)$
signature (2p, 2q) $U(p,q)$ and $SU(p,q)$
signature (4p,4q) $SP(1)SP(p,q)$ $SP(p,q)$
signature (n,n) $SO(n,\mathbb{C})$
signature (2n,2n) $SO(n, \mathbb{H})$

$G_2^\mathbb{C}$ $G_2$ and $G_2^2$ signatures (7,7), (7,0) and (4,3)

$Spin(7,\mathbb{C})$ $Spin(7)$ and Spin(4,3) signatures (8,8) (8,0) and (4,4).

you of course are interested in signature (p,1)

The point is that irreducible holonomies are the ones that act transitively on the unit sphere in the tangent space. And this seems to me as the problem you are asking for.

[2] Proc of Symp in pure mathematics 54 (1993) part 2. On the Holonomy of Lorentzian manifolds. Bergery and Ikemakhen

1

I couldn't write this as a comment and I might be off base, but in Berger's paper Theorem 6 he proves that the list of real Lie groups acting transitively on the quadric $$x_1^2 + \dots + x_{n-h}^2 - x^2_{n-h+1} - \dots - x^2_{n} = 1$$ induced from a linear action on $\mathbb{R}^n$ are modulo a finite number of exceptions
1) $SO(n-h,h)$ $n \geq 2$

2) $T^1 \times SU(m/2-h/2,h/2)$, $SU(m/2-h/2,h/2)$ for some $m$ $m/2 \geq 2$

3) $SP(1) \times SP(m/4-h/2,h/2)$, $T^1 \times SP(m/4-h/2,h/2)$, $SP(m/4-h/2,h/2)$ $m/4 \geq 2$