In reading the text Multi-View Geometry for Computer vision, after the notion of a conic is introduced with its corresponding conic coefficient matrix, an example is given on page 32 stating
The conic $C= lm^T + ml^T$ is composed of two lines $l$ and $m$. Points on $l$ satisfy $l^T x = 0$, and are on the conic since $x^TCx = (x^T l)(m^Tx) + (x^Tm)(l^Tx) = 0$. Similarly, points satisfying $m^Tx = 0$ also satisfy $x^TCx=0$.
I'm not sure what this notation means. I'm guessing that $C$ is the sum of two outer products, since otherwise it would be a sum of scalars, and then $x^TCx$ would be a scalar multiple of the squared norm of $x$.
If it is indeed a sum of outer products, then is it generally true that for an outer product $l \bigotimes m^T$ that $x^T l \bigotimes m^T x = (x^T l)(m^Tx)$ ? (where the latter expression is a product of dot products)
We are given column vectors $$ x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad l = \begin{pmatrix} l_1 \\ l_2 \\ l_3 \end{pmatrix}, \quad m = \begin{pmatrix} m_1 \\ m_2 \\ m_3 \end{pmatrix}.\tag{1} $$ Note that $$ l\,m^T := \begin{pmatrix} l_1m_1 & l_1m_2 & l_1m_3 \\ l_2m_1 & l_2m_2 & l_2m_3 \\l_3m_1 & l_3m_2 & l_3m_3 \end{pmatrix} \tag{2} $$ is a rank $1$ matrix.
The equation $$ l^T x := l_1x_1 + l_2x_2 + l_3x_3 = 0 \tag{3} $$ is the equation representing points $\,x\,$ on the line $\,l.\,$
The equation $$ m^T x := m_1x_1 + m_2x_2 + m_3x_3 = 0 \tag{4} $$ is the equation representing points $\,x\,$ on the line $\,m.\,$
Define the matrix $$ C := lm^T + ml^T = \begin{pmatrix} 2l_1m_1 & l_1m_2+l_2m_1 & l_1m_3+l_3m_1 \\ l_1m_2+l_2m_1 & 2l_2m_2 & l_2m_3+l_3m_2 \\ l_1m_3+l_3m_1 & l_2m_3+l_3m_2 & 2l_3m_3 \end{pmatrix}. \tag{5} $$
This rank $2$ matrix represents a degenerate conic which consists of the two lines $\,l\,$ and $\,m.\,$ The reason is what you mentioned in your question. We have $$ x^TCx = (x^Tl)(m^Tx) + (x^Tm)(l^Tx) = 2(l^Tx)(m^Tx)\tag{6} $$ since $\, l^Tx = x^Tl \,$ and $\, m^Tx = x^Tm \,$ are both scalars. Their product is zero precisely when either of them are zero and each factor is the equation of a line. Hence, this proves the claim about $\,C\,$ representing two lines.