Matrix inconsistency I've been looking into the following deriviation of a parameterization of a green assets Markowitz problem. One author completes the following step in a calculation that I can not follow.
\begin{align}
a&=(1-\lambda)\frac{d}{a^2}\left[ \frac{1}{\eta^2} \left(\underbrace{ I_N-\frac{1}{\eta^2/\sigma^2+N} \mathbf{1}_{N\times 1}\mathbf{1}_{N\times 1}' }_{Y}\right)\right]g\\
&=(1-\lambda)\frac{d}{a^2\eta^2}g
\end{align}
where $I_N$ is the identity matrix in N dimensions; $\mathbf{1}$ is a matrix of all ones; $\lambda, \eta, d, a$ are scalars; g is an $N\times1$ matrix.
The authors mention $\eta^2\approx(0.7/0.3)\sigma^2$, however this feel irrelevant in this context.
The above implies that the entire paranthesised part is $Y=1$? And I do not really understand why? I would be super grateful if anyone could point me in the right direction.
EDIT 1: Additional information
In the list of assumptions these two were shortly mentioned:
$$
(\mathbf{1}_{N\times 1})'g=0, g'g =1 
$$

The key point is that $Yg = g$. To see this, note that
$$
Yg = [I_N - K \cdot \mathbf 1 \mathbf 1^T]g = 
I_N g - K \cdot \mathbf 1 (\mathbf 1^T g) = g - 0 = g.
$$
